Projection Pursuit Regression

Description

Fit a projection pursuit regression model.

Usage

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ppr(x, ...)

## S3 method for class 'formula'
ppr(formula, data, weights, subset, na.action,
    contrasts = NULL, ..., model = FALSE)

## Default S3 method:
ppr(x, y, weights = rep(1, n),
    ww = rep(1, q), nterms, max.terms = nterms, optlevel = 2,
    sm.method = c("supsmu", "spline", "gcvspline"),
    bass = 0, span = 0, df = 5, gcvpen = 1, ...)

Arguments

formula

a formula specifying one or more numeric response variables and the explanatory variables.

x

numeric matrix of explanatory variables. Rows represent observations, and columns represent variables. Missing values are not accepted.

y

numeric matrix of response variables. Rows represent observations, and columns represent variables. Missing values are not accepted.

nterms

number of terms to include in the final model.

data

a data frame (or similar: see model.frame) from which variables specified in formula are preferentially to be taken.

weights

a vector of weights w_i for each case.

ww

a vector of weights for each response, so the fit criterion is the sum over case i and responses j of w_i ww_j (y_ij - fit_ij)^2 divided by the sum of w_i.

subset

an index vector specifying the cases to be used in the training sample. (NOTE: If given, this argument must be named.)

na.action

a function to specify the action to be taken if NAs are found. The default action is given by getOption("na.action"). (NOTE: If given, this argument must be named.)

contrasts

the contrasts to be used when any factor explanatory variables are coded.

max.terms

maximum number of terms to choose from when building the model.

optlevel

integer from 0 to 3 which determines the thoroughness of an optimization routine in the SMART program. See the ‘Details’ section.

sm.method

the method used for smoothing the ridge functions. The default is to use Friedman's super smoother supsmu. The alternatives are to use the smoothing spline code underlying smooth.spline, either with a specified (equivalent) degrees of freedom for each ridge functions, or to allow the smoothness to be chosen by GCV.

Can be abbreviated.

bass

super smoother bass tone control used with automatic span selection (see supsmu); the range of values is 0 to 10, with larger values resulting in increased smoothing.

span

super smoother span control (see supsmu). The default, 0, results in automatic span selection by local cross validation. span can also take a value in (0, 1].

df

if sm.method is "spline" specifies the smoothness of each ridge term via the requested equivalent degrees of freedom.

gcvpen

if sm.method is "gcvspline" this is the penalty used in the GCV selection for each degree of freedom used.

...

arguments to be passed to or from other methods.

model

logical. If true, the model frame is returned.

Details

The basic method is given by Friedman (1984), and is essentially the same code used by S-PLUS's ppreg. This code is extremely sensitive to the compiler used.

The algorithm first adds up to max.terms ridge terms one at a time; it will use less if it is unable to find a term to add that makes sufficient difference. It then removes the least important term at each step until nterms terms are left.

The levels of optimization (argument optlevel) differ in how thoroughly the models are refitted during this process. At level 0 the existing ridge terms are not refitted. At level 1 the projection directions are not refitted, but the ridge functions and the regression coefficients are. Levels 2 and 3 refit all the terms and are equivalent for one response; level 3 is more careful to re-balance the contributions from each regressor at each step and so is a little less likely to converge to a saddle point of the sum of squares criterion.

Value

A list with the following components, many of which are for use by the method functions.

call

the matched call

p

the number of explanatory variables (after any coding)

q

the number of response variables

mu

the argument nterms

ml

the argument max.terms

gof

the overall residual (weighted) sum of squares for the selected model

gofn

the overall residual (weighted) sum of squares against the number of terms, up to max.terms. Will be invalid (and zero) for less than nterms.

df

the argument df

edf

if sm.method is "spline" or "gcvspline" the equivalent number of degrees of freedom for each ridge term used.

xnames

the names of the explanatory variables

ynames

the names of the response variables

alpha

a matrix of the projection directions, with a column for each ridge term

beta

a matrix of the coefficients applied for each response to the ridge terms: the rows are the responses and the columns the ridge terms

yb

the weighted means of each response

ys

the overall scale factor used: internally the responses are divided by ys to have unit total weighted sum of squares.

fitted.values

the fitted values, as a matrix if q > 1.

residuals

the residuals, as a matrix if q > 1.

smod

internal work array, which includes the ridge functions evaluated at the training set points.

model

(only if model = TRUE) the model frame.

Source

Friedman (1984): converted to double precision and added interface to smoothing splines by B. D. Ripley, originally for the MASS package.

References

Friedman, J. H. and Stuetzle, W. (1981) Projection pursuit regression. Journal of the American Statistical Association, 76, 817–823.

Friedman, J. H. (1984) SMART User's Guide. Laboratory for Computational Statistics, Stanford University Technical Report No. 1.

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Springer.

See Also

plot.ppr, supsmu, smooth.spline

Examples

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require(graphics)

# Note: your numerical values may differ
attach(rock)
area1 <- area/10000; peri1 <- peri/10000
rock.ppr <- ppr(log(perm) ~ area1 + peri1 + shape,
                data = rock, nterms = 2, max.terms = 5)
rock.ppr
# Call:
# ppr.formula(formula = log(perm) ~ area1 + peri1 + shape, data = rock,
#     nterms = 2, max.terms = 5)
#
# Goodness of fit:
#  2 terms  3 terms  4 terms  5 terms
# 8.737806 5.289517 4.745799 4.490378

summary(rock.ppr)
# .....  (same as above)
# .....
#
# Projection direction vectors:
#       term 1      term 2
# area1  0.34357179  0.37071027
# peri1 -0.93781471 -0.61923542
# shape  0.04961846  0.69218595
#
# Coefficients of ridge terms:
#    term 1    term 2
# 1.6079271 0.5460971

par(mfrow = c(3,2))   # maybe: , pty = "s")
plot(rock.ppr, main = "ppr(log(perm)~ ., nterms=2, max.terms=5)")
plot(update(rock.ppr, bass = 5), main = "update(..., bass = 5)")
plot(update(rock.ppr, sm.method = "gcv", gcvpen = 2),
     main = "update(..., sm.method=\"gcv\", gcvpen=2)")
cbind(perm = rock$perm, prediction = round(exp(predict(rock.ppr)), 1))
detach()

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