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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. [
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.▼
▲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>
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As a tool for data analysis, the ACE procedure provides graphical output to indicate a need for transformations as well as to guide in their choice. If a particular plot suggests a familiar functional form for a transformation, then the data can be pre-transformed using this functional form and the ACE algorithm can be rerun.
Wang suggests that the [[Power transform|Box-Cox transform]], a parametric approach, is a special case of ACE.<ref>Wang, D., Murphy, M. 2005. Identifying Nonlinear Relationships in Regression using the ACE Algorithm. Journal of Applied Statistics. 32(3) 243-258.</ref>
As with any regression procedure, a high degree of association between predictor variables can sometimes cause the individual transformation estimates to be highly variable, even though the complete model is reasonably stable. When this is suspected, running the algorithm on randomly selected subsets of the data, or on [[Bootstrapping (statistics)|bootstrap samples]] can assist▼
== Limitations ==
▲As with any regression procedure, a high degree of association between predictor variables can sometimes cause the individual transformation estimates to be highly variable, even though the complete model is reasonably stable. When this is suspected, running the algorithm on randomly selected subsets of the data, or on [[Bootstrapping (statistics)|bootstrap samples]] can assist in assessing the variability.
ACE has shown some sensitivity to the order of the predictor variables and extreme outliers.<ref>De Veaux, R. 1990. Finding Transformations for Regression Using the ACE Algorithm. Sociological Methods and Research 18(2-3) 327-359.</ref> Long tailed distributions can lead to the above mentioned instability.
In real world applications one can never be sure that all relevant variables are observed and ACE will always recommend a transform. Thus the recommended transforms can be symptoms of this problem rather than what ACE is trying to solve.<ref>Pregibon, D., Vardi, Y. 1985. Estimating Optimal Transformations for Multiple Regression and Correlation: Comment. Journal of the American Statistical Association. 80(391) 598-601</ref>
== References ==
{{reflist}}
* [[File:PD-icon.svg|15px|link=|alt=]] ''This draft contains quotations from [
[[Category:Nonparametric regression]]
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