Alternating conditional expectations: Difference between revisions

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In [[statistics]], '''Alternating Conditional Expectations (ACE)''' is a [[nonparametric statistics|nonparametric]] [[algorithm]] used in [[regression analysis]] to find the optimal transformations for both the outcome ([[response variable|response]]) variable and the input (predictor) variables.<ref>Breiman, L. and Friedman, J. H. [http://apps.dtic.mil/dtic/tr/fulltext/u2/a123908.pdf Estimating optimal transformations for multiple regression and correlation]. J. Am. Stat. Assoc., 80(391):580–598, September 1985b. {{PD-notice}}</ref>

For example, in a model that tries to predict house prices based on size and ___location, ACE helps by figuring out if, for instance, transforming the size (maybe taking the [[square root]] or logarithm) or the ___location (perhaps grouping locations into categories) would make the relationship easier to model and lead to better predictions. The algorithm iteratively adjusts these transformations until it finds the ones that maximize the [[predictive power]] of the regression model.

==Introduction==

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Let <math>Y,X_1,\dots,X_p</math> be [[Random variable|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 zero-mean functions and with these [[Transformation (function)|transformation functions]], the fraction of variance of <math>\theta(Y)</math> not explained is

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

# Hold <math>\varphi_1(X_1),\dots,\varphi_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)\Bigg|Y\right]</math>