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ace

Alternating Conditional Expectations

Description

Uses the alternating conditional expectations algorithm to find the transformations of y and x that maximise the proportion of variation in y explained by x. When x is a matrix, it is transformed so that its columns are equally weighted when predicting y.

Usage

ace(x, y, wt = rep(1, nrow(x)), cat = NULL, mon = NULL, lin = NULL,
   circ = NULL, delrsq = 0.01)

Arguments

`x`	a matrix containing the independent variables.
`y`	a vector containing the response variable.
`wt`	an optional vector of weights.
`cat`	an optional integer vector specifying which variables assume categorical values. Positive values in `cat` refer to columns of the `x` matrix and zero to the response variable. Variables must be numeric, so a character variable should first be transformed with as.numeric() and then specified as categorical.
`mon`	an optional integer vector specifying which variables are to be transformed by monotone transformations. Positive values in `mon` refer to columns of the `x` matrix and zero to the response variable.
`lin`	an optional integer vector specifying which variables are to be transformed by linear transformations. Positive values in `lin` refer to columns of the `x` matrix and zero to the response variable.
`circ`	an integer vector specifying which variables assume circular (periodic) values. Positive values in `circ` refer to columns of the `x` matrix and zero to the response variable.
`delrsq`	termination threshold. Iteration stops when R-squared changes by less than `delrsq` in 3 consecutive iterations (default 0.01).

Value

A structure with the following components:

`x`	the input x matrix.
`y`	the input y vector.
`tx`	the transformed x values.
`ty`	the transformed y values.
`rsq`	the multiple R-squared value for the transformed values.
`l`	the codes for cat, mon, ...
`m`	not used in this version of ace

References

Breiman and Friedman, Journal of the American Statistical Association (September, 1985).

The R code is adapted from S code for avas() by Tibshirani, in the Statlib S archive; the FORTRAN is a double-precision version of FORTRAN code by Friedman and Spector in the Statlib general archive.

Examples

TWOPI <- 8*atan(1)
x <- runif(200,0,TWOPI)
y <- exp(sin(x)+rnorm(200)/2)
a <- ace(x,y)
par(mfrow=c(3,1))
plot(a$y,a$ty)  # view the response transformation
plot(a$x,a$tx)  # view the carrier transformation
plot(a$tx,a$ty) # examine the linearity of the fitted model

# example when x is a matrix
X1 <- 1:10
X2 <- X1^2
X <- cbind(X1,X2)
Y <- 3*X1+X2
a1 <- ace(X,Y)
plot(rowSums(a1$tx),a1$y)
(lm(a1$y ~ a1$tx)) # shows that the colums of X are equally weighted

# From D. Wang and M. Murphy (2005), Identifying nonlinear relationships
# regression using the ACE algorithm.  Journal of Applied Statistics,
# 32, 243-258.
X1 <- runif(100)*2-1
X2 <- runif(100)*2-1
X3 <- runif(100)*2-1
X4 <- runif(100)*2-1

# Original equation of Y:
Y <- log(4 + sin(3*X1) + abs(X2) + X3^2 + X4 + .1*rnorm(100))

# Transformed version so that Y, after transformation, is a
# linear function of transforms of the X variables:
# exp(Y) = 4 + sin(3*X1) + abs(X2) + X3^2 + X4

a1 <- ace(cbind(X1,X2,X3,X4),Y)

# For each variable, show its transform as a function of
# the original variable and the of the transform that created it,
# showing that the transform is recovered.
par(mfrow=c(2,1))

plot(X1,a1$tx[,1])
plot(sin(3*X1),a1$tx[,1])

plot(X2,a1$tx[,2])
plot(abs(X2),a1$tx[,2])

plot(X3,a1$tx[,3])
plot(X3^2,a1$tx[,3])

plot(X4,a1$tx[,4])
plot(X4,a1$tx[,4])

plot(Y,a1$ty)
plot(exp(Y),a1$ty)

acepack

ACE and AVAS for Selecting Multiple Regression Transformations

v1.4.1

MIT + file LICENSE

Authors

Phil Spector, Jerome Friedman, Robert Tibshirani, Thomas Lumley, Shawn Garbett, Jonathan Baron

Initial release

ace