Alternating conditional expectations: Difference between revisions

Revision as of 11:41, 11 July 2022

Revision as of 17:57, 3 February 2024

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==Introduction==

In [[statistics]], a nonlinear transformation of variables is commonly used in practice in regression problems. Alternating conditional expectations (ACE) is one of the methods to find those transformations that produce the best fitting [[additive model]]. Knowledge of such transformations aids in the interpretation and understanding of the relationship between the response and predictors.

ACE transformtransforms the response variable <math>Y</math> and its predictor variables, <math>X_i</math> to minimize the [[Fraction of variance unexplained|fraction of variance not explained]]. The transformation is nonlinear and is iteratively obtained from data in an iterative way.

== Mathematical description ==

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where <math>\rho</math> is [[Pearson correlation coefficient]]. <math> \rho^*(X, Y)</math> is known as the maximal correlation between <math>X</math> and <math>Y</math>. It can be used as a general measure of dependence.

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

== Software implementation ==

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== Discussion ==

The ACE algorithm provides a fully automated method for estimating optimal transformations in [[Regression analysis|multiple regression]]. It also provides a method for estimating the maximal correlation between random variables. Since the process of iteration usually terminates in a limited number of runs, the time complexity of the algorithm is <math>O(np)</math> where <math>n</math> is the number of samples. The algorithm is reasonably computer efficient.

A strong advantage of the ACE procedure is the ability to incorporate variables of quite different typetypes in terms of the set of values they can assume. The transformation functions <math>\theta(y), \varphi_i(x_i)</math> assume values on the real line. Their arguments can, however, assume values on any set. For example, ordered real and unordered [[Categorical variable|categorical variables]] can be incorporated in the same regression equation. Variables of mixed type are admissible.

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.