Alternating conditional expectations: Difference between revisions

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== Mathematical description ==

Let <math>Y,X_1,\dots,X_p</math> be random variables. We use <math>X_1,\dots,X_p</math> to predict <math>Y</math>. Suppose <math>\theta(Y),\varphi_1(X_1),\dots,\varphi_p(X_p)</math> are mean-zero-mean functions and with these transformation functions, the fraction of variance of <math>\theta(Y)</math> not explained is

: <math> e^2(\theta,\varphi_1,\dots,\varphi_p)=\frac{E[\theta(Y)-\sum_{i=1}^p \varphi_i(X_i)]^2}{E\theta^2(Y)}</math>

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: