avas: Additivity and variance stabilization for regression

Description

Estimate transformations of x and y such that the regression of y on x is approximately linear with constant variance

Usage

avas(...)
# S3 method for default
avas(
  x,
  y,
  wt = NULL,
  cat = NULL,
  mon = NULL,
  lin = NULL,
  circ = NULL,
  delrsq = 0.01,
  yspan = 0,
  control = NULL,
  ...
)
# S3 method for formula
avas(
  formula,
  data = NULL,
  subset = NULL,
  na.action = getOption("na.action"),
  ...
)
# S3 method for avas
summary(object, ...)
# S3 method for avas
print(x, ..., digits = 4)
# S3 method for avas
plot(
  x,
  ...,
  which = 1:(x$p + 1),
  caption = c(list("Response Y AVAS Transformation"), as.list(paste("Carrier",
    rownames(x$x), "AVAS Transformation"))),
  xlab = "Original",
  ylab = "Transformed",
  ask = prod(par("mfcol")) < length(which) && dev.interactive()
)

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 avas
yspan: span used for smoothing the variance
iters: iteration number and rsq for that iteration
niters: number of iterations used

Arguments

...: additional arguments which go ignored for avas call. Included for S3 dispatch consistency. They are utilized when using print as they get passed to cat. Also when plotting an ace object they are passed to plot.
x: 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
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: numeric(1); Termination threshold for iteration. Stops when R-squared changes by less than delrsq in 3 consecutive iterations (default 0.01).
yspan: yspan Optional window size parameter for smoothing the variance. Range is \([0,1]\). Default is 0 (cross validated choice). .5 is a reasonable alternative to try.
control: named list; control parameters to set. Documented at set_control.
formula: formula; an object of class "formula": a symbolic description of the model to be smoothed.
data: an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which ace is called.
subset: an optional vector specifying a subset of observations to be used in the fitting process. Only used when a formula is specified.
na.action: a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The ‘factory-fresh’ default is na.omit. Another possible value is NULL, no action. Value na.exclude can be useful.
object: an S3 ace object
digits: rounding digits for summary/print
which: when plotting an ace object which plots to produce.
caption: a list of captions for a plot.
xlab: the x-axis label when plotting.
ylab: the y-axis label when plotting.
ask: when plotting should the terminal be asked for input between plots.

References

Rob Tibshirani (1987), ``Estimating optimal transformations for regression''. Journal of the American Statistical Association 83, 394ff.

Examples

Run this code


TWOPI <- 8*atan(1)
x <- runif(200,0,TWOPI)
y <- exp(sin(x)+rnorm(200)/2)
a <- avas(x,y)
plot(a) # View response and carrier transformations
plot(a$tx,a$ty) # examine the linearity of the fitted model

# From D. Wang and M. Murphy (2005), Identifying nonlinear relationships
# regression using the ACE algorithm.  Journal of Applied Statistics,
# 32, 243-258, adapted for avas.
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 <- avas(cbind(X1,X2,X3,X4),Y)

par(mfrow=c(2,1))

# 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.
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)

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