2D Gaussian distribution: remark on confidence rectangles and ellipses with correlated variables (Matlab example)

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 08:29

Summary:

This video reviews the main ideas of the video [cfrc1EN], and raises the question of whether or not the confidence region computations in that video would still be correct with correlated Gaussian variables (non-diagonal covariance matrix).

Briefly, it is proposed that the ellipsoid will be correctly computed (although it will not be aligned with the axes, the relationship of the geometry of said ellipsoid with the eigenvalues and eigenvectors of the variance-covariance matrix is discussed). Also the marginal confidence intervals will be correct (the marginal confidence intervals of a 2D normal are a 1D normal).

However, the ’rectangles’ that were obtained by multiplying the probabilities of the confidence intervals would NOT give a correct probability result as marginals were not independent variables. To do this right, the rectangles should be aligned with the axes of the ellipse (principal components). This fact of transforming the variables of a multidimensional normal (rotate) so that the covariance matrix is diagonal (confidence ellipsoids aligned with the axes) is what motivates what is called principal component analysis in multivariate statistics.

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