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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
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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
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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:
256:, 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
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1953:. 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
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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.
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A strong advantage of the ACE procedure is the ability to incorporate variables of quite different types in terms of the set of values they can assume. The transformation functions
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Estimating
Optimal Transformations For Multiple Regression And Correlation By Leo Breiman And Jerome Freidman. Technical Report No. 9 July 1982
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260:. Knowledge of such transformations aids in the interpretation and understanding of the relationship between the response and predictors.
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In the bivariate case, the ACE algorithm can also be regarded as a method for estimating the maximal correlation between two variables.
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assume values on the real line. Their arguments can, however, assume values on any set. For example, ordered real and unordered
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1535:{\displaystyle \rho ^{*}(X,Y)=\rho ^{*}(\theta ^{*},\varphi ^{*})=\max _{\theta ,\varphi }\rho (\theta (Y),\varphi (X))}
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759:{\displaystyle e^{2}(\theta ,\varphi _{1},\dots ,\varphi _{p})={\frac {\mathbb {E} \left^{2}}{\mathbb {E} }}}
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The ACE algorithm provides a fully automated method for estimating optimal transformations in
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can be incorporated in the same regression equation. Variables of mixed type are admissible.
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Estimating optimal transformations for multiple regression and correlation
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is the number of samples. The algorithm is reasonably computer efficient.
530:{\displaystyle \theta (Y),\varphi _{1}(X_{1}),\dots ,\varphi _{p}(X_{p})}
314:. The transformation is nonlinear and is iteratively obtained from data.
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which implements ACE algorithm. The following example shows its usage:
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This article incorporates text from this source, which is in the
845:{\displaystyle \varphi _{1}(X_{1}),\dots ,\varphi _{p}(X_{p})}
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2102:. J. Am. Stat. Assoc., 80(391):580–598, September 1985b.
1271:{\displaystyle {\tilde {\varphi }}_{k}=\mathbb {E} \left}
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1651:. It can be used as a general measure of dependence.
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may be too technical for most readers to understand
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983:{\displaystyle \theta _{1}(Y)=\mathbb {E} \left}
236:to find the optimal transformations between the
2059:{\displaystyle \theta (y),\varphi _{i}(x_{i})}
1367:{\displaystyle \theta ^{*}(Y),\varphi ^{*}(X)}
1611:is known as the maximal correlation between
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89:introducing citations to additional sources
1939:# examine the linearity of the fitted model
50:Learn how and when to remove these messages
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213:Learn how and when to remove this message
195:Learn how and when to remove this message
179:, without removing the technical details.
79:Relevant discussion may be found on the
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537:are zero-mean functions and with these
100:"Alternating conditional expectations"
263:ACE transforms the response variable
177:make it understandable to non-experts
7:
2126:This draft contains quotations from
1280:Iterate the above three steps until
367:{\displaystyle Y,X_{1},\dots ,X_{p}}
226:Alternating conditional expectations
1087:{\displaystyle \varphi _{i}(X_{i})}
1873:# view the response transformation
417:{\displaystyle X_{1},\dots ,X_{p}}
312:fraction of variance not explained
14:
1906:# view the carrier transformation
31:This article has multiple issues.
2130:, which is in the public domain.
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2098:Breiman, L. and Friedman, J. H.
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72:relies largely or entirely on a
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1567:Pearson correlation coefficient
39:or discuss these issues on the
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283:and its predictor variables,
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240:and predictor variables in
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1116:{\displaystyle \theta (Y)}
563:{\displaystyle \theta (Y)}
2145:Nonparametric regression
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539:transformation functions
318:Mathematical description
1658:Software implementation
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1150:and the solution is::
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2068:categorical variables
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102: –
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248:Introduction
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33:Please help
30:
424:to predict
185:August 2018
141:August 2018
2086:References
1945:Discussion
1664:R language
1400:satisfies
992:Normalize
444:. Suppose
254:statistics
111:newspapers
36:improve it
2032:φ
2016:θ
1582:∗
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1518:φ
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1222:φ
1213:≠
1206:∑
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1060:φ
1031:For each
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378:. We use
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234:algorithm
81:talk page
42:talk page
2139:Category
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132:JSTOR
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