And the volume of the ellipsoid is proportional to the product of the eigenvalues of ( X T X) −1, that is, it is proportional to det. Moreover, large correlations among the columns of X (collinearity) will be translated into large differences between the largest and the smallest eigenvalues of X and, hence, into very different ellipsoid axis lengths (a narrow and long ellipsoid). Next, the half-length of the axis is dλ a′ −1/2 = dλ a 1/2, which means that a large ( X T X) −1 will be translated into large eigenvalues, and, in turn, into a large ellipsoid. Hence, the axes of the ellipsoid are oriented in the direction of the eigenvectors of ( X T X) −1. First, we must note that X T X and ( X T X) −1 have the same eigenvectors, and that their eigenvalues are related as λ a′ = λ a −1. This is translated into the confidence ellipsoid as follows. Large values in ( X T X) −1 imply large variances and covariances. The importance of this matrix is better seen in terms of the dispersion matrix ( X T X) −1, which is used for calculating the variances and covariances of the coefficients. It does not store any personal data.( β − b ) T X T X ( β − b ) = K s 2 F K, ( I − K ), 1 − αĪnd making A = X T X, d 2 = Ks 2 F K,( I − K),1 − α, it is seen that Equation (49) defines an ellipsoid whose shape and volume depend on the design matrix X. The cookie is set by the GDPR Cookie Consent plugin and is used to store whether or not user has consented to the use of cookies. The cookie is used to store the user consent for the cookies in the category "Performance".
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POSITIVE DEFINITE MATRIX CODE
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