Robust Boosting for Robust Loss Functions

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Description

MM (majorization/minimization) algorithm based gradient boosting for optimizing nonconvex robust loss functions with componentwise linear, smoothing splines, tree models as base learners.

Usage

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rbst(x, y, cost = 0.5, rfamily = c("tgaussian", "thuber","thinge", "tbinom", "binomd", 
"texpo", "tpoisson", "clossR", "closs", "gloss", "qloss"), ctrl=bst_control(), 
control.tree=list(maxdepth = 1), learner=c("ls","sm","tree"),del=1e-10)

Arguments

x

a data frame containing the variables in the model.

y

vector of responses. y must be in {1, -1}.

cost

price to pay for false positive, 0 < cost < 1; price of false negative is 1-cost.

rfamily

family = "tgaussian" for truncated square error loss, "thinge" for truncated hinge loss, "tbinom" for truncated logistic loss, "binomd" for logistic difference loss, "tpoisson" for truncated Poisson loss.

ctrl

an object of class bst_control.

control.tree

control parameters of rpart.

learner

a character specifying the component-wise base learner to be used: ls linear models, sm smoothing splines, tree regression trees.

del

convergency critera

Details

An MM algorithm operates by creating a convex surrogate function that majorizes the nonconvex objective function. When the surrogate function is minimized with gradient boosting algorithm, the desired objective function is decreased. The MM algorithm contains difference of convex (DC) algorithm for rfamily=c("tgaussian", "thuber","thinge", "tbinom", "binomd", "texpo", "tpoisson") and quadratic majorization boosting algorithm (QMBA) for rfamily=c("clossR", "closs", "gloss", "qloss").

Value

An object of class bst with print, coef, plot and predict methods are available for linear models. For nonlinear models, methods print and predict are available.

x, y, cost, rfamily, learner, control.tree, maxdepth

These are input variables and parameters

ctrl

the input ctrl with possible updated fk if family="tgaussian", "thingeDC", "tbinomDC", "binomdDC", "tpoisson"

yhat

predicted function estimates

ens

a list of length mstop. Each element is a fitted model to the psedo residuals, defined as negative gradient of loss function at the current estimated function

ml.fit

the last element of ens

ensemble

a vector of length mstop. Each element is the variable selected in each boosting step when applicable

xselect

selected variables in mstop

coef

estimated coefficients in mstop

Author(s)

Zhu Wang

See Also

cv.bst for cross-validated stopping iteration. Furthermore see bst_control

Examples

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x <- matrix(rnorm(100*5),ncol=5)
c <- 2*x[,1]
p <- exp(c)/(exp(c)+exp(-c))
y <- rbinom(100,1,p)
y[y != 1] <- -1
y[1:10] <- -y[1:10]
x <- as.data.frame(x)
dat.m <- bst(x, y, ctrl = bst_control(mstop=50), family = "hinge", learner = "ls")
predict(dat.m)
dat.m1 <- bst(x, y, ctrl = bst_control(twinboost=TRUE, 
coefir=coef(dat.m), xselect.init = dat.m$xselect, mstop=50))
dat.m2 <- rbst(x, y, ctrl = bst_control(mstop=50, s=0, trace=TRUE), 
rfamily = "thinge", learner = "ls")
predict(dat.m2)