Multivariate Gaussianization for Data Processing
Multivariate Gaussianization for Data Processing
Multivariate Gaussianization for Data Processing
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Intro Iterative <strong>Gaussianization</strong> Experiments ConclusionsMarginal (univariate) <strong>Gaussianization</strong>Marginal <strong>Gaussianization</strong> is trivial [Friedman87]<strong>Gaussianization</strong> in each dimension, Ψ i (k), can be decomposed into twoconsecutive equalization trans<strong>for</strong>ms:1 Marginal uni<strong>for</strong>mization, U(k) i , based on the cdf of the marginal PDF,2 <strong>Gaussianization</strong> of a uni<strong>for</strong>m variable, G(u), based on the inverse of the cdfof a univariate Gaussian: Ψ i (k) = G ⊙ Ui (k)whereu = U i (k)(x (k)i) =G −1 (x i ) =∫ x(k)i−∞∫ xi−∞p i (x ′(k)i) dx ′(k)ig(x i ′ ) dx i′and g(x i ) is just a univariate Gaussian.0.011150.5p i(x i)0.0080.0060.004u = U i (x i)0.80.60.4p(u)0.80.60.4G(u) = Ψ i (x i)0p i(Ψ i (x i))0.40.30.20.0020.20.20.100 100 200 300x i00 100 200 300x i00 0.5 1u−50 0.5 1u0−5 0 5G(u) = Ψ i (x i)
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