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To check the linearity of the model described in Section 2, we first choose the coordinate system of the hours having the extreme correlation values with employment levels.
The absolute value of the partial correlation of a 3×3 extreme correlation matrix of the form (2.1) is always one.
The bridge is modeled using different types of systems with the consideration of two extreme correlation cases among the failure modes of the girders.
As observed by Devroye and Letac (2010), any valid rank correlation matrix can be written as a finite convex combination of extreme correlation matrices, also called extreme points (Theorem of Carathéodory (1911) and Steinitz (1914)).
Advocating the Theorem of Carathéodory, which states that any valid correlation matrix can be written as a finite convex combination of extreme correlation matrices, the generation of 3-universal copulas follows.
Based on the trivariate pair-copula construction for the bivariate linear circular copula by Perlman and Wellner (Symmetry 3 574-99, 2011) and the Theorem of Carathéodory, which states that any valid correlation matrix is a finite convex combination of extreme correlation matrices, we generate a class of closed-form analytical 3-universal copulas.
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The components were chosen based on their high concentrations in tested oils and extreme correlations between amounts and antioxidant properties.
However, when put in a genomic context, these correlations are not exceptional: out of 585,379 SNPs with minor allele frequency >0.05 genotyped in the same individuals by Li et al. [ 32], 8.6% had more extreme correlations with absolute latitude, and 14% after correcting for distance to East Africa.
Even with all observation types, there is extreme parameter correlation between porosity and hydraulic conductivity and between the sorption rate and in-solution inactivation.
The most extreme genetic correlation between healthy and affected birds was 0.45.
In this cross-comparison analysis, the extreme low correlation coefficients might be the consequence of the above-suggested loss of intensity levels.
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