Alternating conditional expectations: Difference between revisions

Revision as of 20:28, 19 June 2018

Revision as of 19:08, 26 June 2018

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The transformation is nonlinear and is obtained from data in an iterative way.

== 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 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>

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In the bivariate case, ACE algorithm can also be regarded as a method for estimating the maximal correlation between two variables.

== Software Implementation ==

The ACE algorithm was developed in the context of known distributions. In practice, data distributions are seldom known and the conditional expectation

should be estimated from data. [[R language]] has a package <kbd>acepack</kbd> which implements ACE algorithm. The following example shows its usage:

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* [[File:PD-icon.svg|15px|link=|alt=]] ''This draft contains quotations from [http://www.dtic.mil/dtic/tr/fulltext/u2/a123908.pdf Estimating Optimal Transformations For Multiple Regression And Correlation By Leo Breiman And Jerome Freidman. Technical Report No. 9 July 1982], which is in the public ___domain.''

[[:Category:Nonparametric regression‎regression]]

{{AFC submission|||ts=20180618031147|u=Zhaofeng-shu33|ns=118}}