Regression Model Linearization
DESCRIPTION:
Performs a form of nonlinear regression which nonparametrically transforms both the y and the x variables to produce an additive model. The transformations are chosen to maximize the correlation between the transformed y and the sum of the transformed x's.
USAGE: 
ace(x, y, wt=<<see below>>, monotone=NULL, linear=NULL,
     categorical=NULL, circular=NULL, tolerance=0.01, n.solution=1)
REQUIRED ARGUMENTS:
x:
vector or matrix containing the explanatory variables. Columns are variables, rows are observations. Missing values are not accepted.
y:
vector containing the response variable. Missing values are not accepted.
OPTIONAL ARGUMENTS:
wt:
vector of weights. The length of wt should be the same as the length of y. By default an unweighted regression is carried out (all weights unity). Missing values are not accepted.
monotone:
integer vector specifying which variables are to be transformed by monotone transformations. Positive values in monotone refer to the columns of the x matrix, 0 refers to y.
linear:
integer vector specifying which variables are to be transformed by linear transformations. Positive values in linear refer to columns of the x matrix and 0 refers to the y variable.
categorical:
integer vector specifying which variables assume categorical (unordered) values. Positive values in categorical refer to columns of the x matrix, 0 to y.
circular:
integer vector specifying which variables assume circular (periodic) values. Positive values in circular refer to columns of the x matrix, 0 to y. All values of a circular variable must lie in the range [0, 1]; if this is not true, a general ordered (default) transformation will be performed.
tolerance:
termination threshold. Iteration stops when the multiple R-squared changes by less than tolerance in 3 consecutive iterations.
n.solution:
desired number of eigen solutions.
VALUE:
a list with the following components:
tx:
an object like x or a list of length n.solution of objects like x (if n.solution is greater than 1) containing the transformed x values.
ty:
a vector or a list of length n.solution of vectors (if n.solution is greater than 1) containing the transformed y values.
rsq:
vector of the multiple R-squared value for the transformed values for each eigen solution.
iterations:
vector of the number of iterations for each eigen solution.
DETAILS:
ace stands for Alternating Conditional Expectation. Transformations are iteratively found using Super Smoother until the correlation fails to increase. The default transformation assumes (for purposes of smoothing) that values of the variable imply an ordering of the variable but no restriction is placed on the transformation.

The data are passed into the Fortran routine in single precision, but key calculations are performed in double precision.

BACKGROUND:
If the first and second solutions have approximately equal (and large) rsq, then any linear combination of the two transformations will have a large rsq as well. Such an occurrence may be due to clustering in the x variables so that an approximate step function is optimal.
REFERENCES:
Breiman, L. and Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation. (with discussion). Journal of the American Statistical Association 80, 580-619.

The chapter "Regression and Smoothing for Continous Response Data" in the S-PLUS Guide to Statistical and Mathematical Analysis.

SEE ALSO:
avas , ppreg , gam .
EXAMPLES: 
x <- runif(200,0,2*pi)
y <- exp(sin(x)+rnorm(200)/2)
exace <- ace(x,y)