Mutual Information and Maximal Correlation as Measures of Dependence

Bell, Caryn N.

doi:10.1214/aoms/1177704583

Cited by 69 publications

(47 citation statements)

References 12 publications

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“…Rényi (1959) enunciated seven properties which should be verified by any dependence measure between two random variables defined over the same probability space. These axioms were discussed and partially modified by Bell (1962), Scheizer and Wolff (1981) and Nelsen (2006), among others.…”

Section: Introductionmentioning

confidence: 99%

Measuring non-linear dependence for two random variables distributed along a curve

Delicado

Smrekar

2008

Stat Comput

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We propose new dependence measures for two real random variables not necessarily linearly related. Covariance and linear correlation are expressed in terms of principal components and are generalized for variables distributed along a curve. Properties of these measures are discussed. The new measures are estimated using principal curves and are computed for simulated and real data sets. Finally, we present several statistical applications for the new dependence measures.

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Section: Introductionmentioning

confidence: 99%

Measuring non-linear dependence for two random variables distributed along a curve

Delicado

Smrekar

2008

Stat Comput

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“…2 (x), ζ 2 (x) = −1/(2σ 2 (x)) corresponding to the canonical statistics t 1 Local strength of dependence The limiting local discrepancy, quantifying the change of the distribution due to a change in x, is given by J 0 (x) = I(x) as x is univariate. Regarding the Normal distributions as a one-parameter exponential family where only changes in μ(x) -though weighted by σ 2 (x) -are reflected,…”

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confidence: 99%

On association in regression: the coefficient of determination revisited

Linde

Tutz

2008

Statistics

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Universal coefficients of determination are investigated which quantify the strength of the relation between a vector of dependent variables Y and a vector of independent covariates X. They are defined as measures of dependence between Y and X through θ(x), with θ(x) parameterizing the conditional distribution of Y given X = x. If θ(x) involves unknown coefficients γ the definition is conditional on γ, and in practice γ, respectively the coefficient of determination has to be estimated. The estimates of quantities we propose generalize R 2 in classical linear regression and are also related to other definitions previously suggested. Our definitions apply to generalized regression models with arbitrary link functions as well as multivariate and nonparametric regression. The definition and use of the proposed coefficients of determination is illustrated for several regression problems with simulated and real data sets.

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“…The mathematical basis for this approach was established earlier by a mathematical proof that such a measure enabled one to reveal dependencies between random variables. 12,13 Biological tissues often exhibit characteristic regular features or ornamental patterns. Therefore, various regions of the tissue image can be statistically well correlated.…”

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confidence: 99%

Visualization of biological texture using correlation coefficient images

et al. 2006

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Abstract. Subsurface structural features of biological tissue are visualized using polarized light images. The technique of Pearson correlation coefficient analysis is used to reduce blurring of these features by unpolarized backscattered light and to visualize the regions of high statistical similarities within the noisy tissue images. It is shown that under certain conditions, such correlation coefficient maps are determined by the textural character of tissues and not by the chosen region of interest, providing information on tissue structure. As an example, the subsurface texture of a demineralized tooth sample is enhanced from a noisy polarized light image. Characterization of biological tissues from intrinsically noisy digital images is currently considered in the frames of several scientific approaches, i.e., pattern recognition, texture analysis, and image processing. [1][2][3][4] Many methods are suggested to increase the contrast, eliminate scattered bright or dark pixels, and filter out noise and certain artifacts. Among the most popular are gray-level run techniques, 5 fast Fourier transforms, 6 and wavelet analysis. 7 None is universal in application, but individual user preference is commonly based on features of the data. Texture analysis aims to characterize the image using a number of parameters that are sensitive to specific structural features of the object and especially to distinguish normal and abnormal biological tissues in the noisy images. For example, a few methods were developed for the detection of clustered microcalcifications from digital mammograms. [8][9][10] Texture enhancement is also required when imaging subsurface tissue structures with a diffusively backscattered polarized beam. Previously it was shown that imaging the degree of polarization can enhance the visibility of the hidden x-ray-induced early fibrosis of mouse skin.11 The analysis of Pearson correlation coefficients was also used to estimate local variations of the directionality and orientation of the structural elements. That work led us to visualize the hidden structures of biological tissues by imaging regions of statistical similarities using the correlation coefficient as the measure of comparison. The mathematical basis for this approach was established earlier by a mathematical proof that such a measure enabled one to reveal dependencies between random variables. 12,13 Biological tissues often exhibit characteristic regular features or ornamental patterns. Therefore, various regions of the tissue image can be statistically well correlated. By choosing a region of interest ͑ROI͒ and comparing it with other regions of the image through the correlation coefficient, it is possible to map the degree of statistical similarities. This mapping can carry valuable comparative information about the structural features of the tissue.Let us consider the main principles of image processing using correlation coefficients. To evaluate the spatial correlation in the raw image data, one has an intensity matrix G. Then two sim...

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Mutual Information and Maximal Correlation as Measures of Dependence

Cited by 69 publications

References 12 publications

Measuring non-linear dependence for two random variables distributed along a curve

Measuring non-linear dependence for two random variables distributed along a curve

On association in regression: the coefficient of determination revisited

Visualization of biological texture using correlation coefficient images

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