Alternating conditional expectations: Difference between revisions

Revision as of 07:33, 8 August 2018

Revision as of 07:34, 8 August 2018

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Generally, the optimal transformations that minimize the unexplained part are difficult to compute directly. As an alternative, ACE is an iterative method to calculate the optimal transformations. The procedure of ACE has the following steps:

# Hold <math>\phi_1(X_1),\dots,\phi_p(X_p)</math> fixed, minimizing <math>e^2</math><!--

-->gives <math>\theta_1(Y)=\mathbb{E}\left[\sum_{i=1}^p \varphi_i(X_i)\bigBigg|Y\right]</math>

# Normalize <math>\theta_1(Y)</math> to unit variance.

# For each <math>k</math>, fix other <math>\varphi_i(X_i)</math> and <math>\theta(Y)</math>, minimizing <math>e^2</math> and the solution is<!--

-->:: <math>\tilde{\varphi}_k = \mathbb{E}\left[\theta(Y)-\sum_{i\neq k} \varphi_i(X_i) \bigBigg| X_k\right]</math>

# Iterate the above three steps until <math>e^2</math> is within error tolerance.