mboost (version 1.0-1)

baselearners: Base learners for Gradient Boosting with Smooth Components

Description

Base learners to be utilized in the formula specification of gamboost().

Usage

bols(x, z = NULL, xname = NULL, zname = NULL)
bbs(x, z = NULL, df = 4, knots = NULL, degree = 3, differences = 2, 
    center = FALSE, xname = NULL, zname = NULL)
bns(x, z = NULL, df = 4, knots = NULL, differences = 2, 
    xname = NULL, zname = NULL)
bss(x, df = 4, xname = NULL)
bspatial(x, y, z = NULL, df = 5, xknots = NULL, yknots = NULL,
         degree = 3, differences = 2, center = FALSE, xname = NULL,
         yname = NULL, zname = NULL)
brandom(x, z = NULL, df = 4, xname = NULL, zname = NULL)

Arguments

x

a vector containing data, either numeric or a factor.

y

a vector containing numeric data

z

an optional vector containing numeric data.

xname

an optional string indicating the name of the variable whose data values are given by the vector x.

yname

an optional string indicating the name of the variable whose data values are given by the vector y.

zname

an optional string indicating the name of the variable whose data values are given by the vector z.

df

trace of the hat matrix for the base learner defining the base learner complexity. Low values of df correspond to a large amount of smoothing and thus to "weaker" base learners. Certain restrictions have to be ke

knots

either the number of (equidistant) interior knots to be used for the regression spline fit or a vector including the positions of the interior knots. If knots=NULL, the interior knots are chosen to be equidis

xknots

knots in x-direction when fitting a bivariate surface with bspatial. See knots for details.

yknots

knots in y-direction when fitting a bivariate surface with bspatial. See knots for details.

degree

degree of the regression spline.

differences

natural number between 1 and 3. If differences = k, k-th-order differences are used as a penalty.

center

If center=TRUE, the corresponding effect is re-parameterized such that the unpenalized part of the fit is substracted and only the deviation effect is fitted. The unpenalized, parametric part has then

Value

• Either a matrix (in case of an ordinary least squares fit) or an object of class basis (in case of a regression or smoothing spline fit) with a dpp function as an additional attribute. The call of dpp returns an object of class basisdpp.

Details

bols refers to linear base learners (ordinary least squares fit), while bbs, bns, and bss refer to penalized regression splines, penalized natural splines, and smoothing splines, respectively. bspatial fits bivariate surfaces and brandom defines random effects base learners. In combination with option z, all base learners can be turned into varying coefficient terms. With bbs, the P-spline approach of Eilers and Marx (1996) is used. bns uses the same penalty and interior knots as bbs, but operates with a constrained natural spline basis instead of an unconstrained B-spline basis. P-splines use a squared k-th-order difference penalty which can be interpreted as an approximation of the integrated squared k-th derivative of the spline. This approximation is only valid if the knots are equidistant, so is not recommended to use non-equidistant knots for bbs and bns. bss refers to a smoothing spline based on the smooth.spline function. bspatial implements bivariate tensor product P-splines for the estimation of either spatial effects (if x and y correspond to coordinates) or interaction surfaces. The penalty term is constructed based on bivariate extensions of the univariate penalties in x and y directions, see Kneib, Hothorn and Tutz (2007) for details. brandom specifies a random effects base learner based on a factor variable x that defines the grouping structure of the data set. For each level of x, a separate random intercept is fitted, where the random effects variance is governed by the specification of the degrees of freedom df. For all base learners except bols, the amount of smoothing is determined by the trace of the hat matrix, as indicated by df. If z is specified as an additional argument, a varying coefficients term is estimated, where z is the interaction variable and the effect modifier is given by either x or x and y. If only x is specified and one of the nonparametric base learners bbs, bns or bss is used, this corresponds to the classical situation of varying coefficients, where the effect of z varies over the domain of x. In case of bspatial as base learner, the effect of z varies with respect to both x and y, i.e. an interaction surface between x and y is specified as effect modifier. For brandom specification of z leads to the estimation of random slopes for covariate z with grouping structure defined by factor x instead of a simple random intercept. For bbs and bspatial, option center requests that the fitted effect is centered around its parametric, unpenalized part. For example, with second order difference penalty, a linear effect of x remains unpenalized by bbs and therefore the degrees of freedom for the base learner have to be larger than 2. To avoid this restriction, option center=TRUE substracts the unpenalized linear effect from the fit, allowing to specify any positive number as df. Note that in this case the linear effect x should generally be specified as an additional base learner bols(x). For bspatial and, for example, second order differences, a linear effect of x (bols(x)), a linear effect of y (bols(y)), and their interaction (bols(x*y)) are substracted from the effect and have to be added seperately to the model equation. More details on centering can be found in Kneib, Hothorn and Tutz (2007) and Fahrmeir, Kneib and Lang (2004).

References

Paul H. C. Eilers and Brian D. Marx (1996), Flexible smoothing with B-splines and penalties. Statistical Science, 11(2), 89-121. Ludwig Fahrmeir, Thomas Kneib and Stefan Lang (2004), Penalized structured additive regression for space-time data: a Bayesian perspective. Statistica Sinica, 14, 731-761. Thomas Kneib, Torsten Hothorn and Gerhard Tutz (2007), Variable selection and model choice in geoadditive regression models. Technical Report No. 3, Institut fuer Statistik, LMU Muenchen. http://epub.ub.uni-muenchen.de/2063/

Examples

x1 <- rnorm(100)
x2 <- rnorm(100) + 0.25*x1
x3 <- as.factor(sample(0:1, 100, replace = TRUE))
y <- 3*sin(x1) + x2^2 + rnorm(100)

knots.x2 <- quantile(x2, c(0.25,0.5,0.75))

spline1 <- bbs(x1,knots=20,df=4)
attributes(spline1)
spline2 <- bns(x2,knots=knots.x2,df=5)
attributes(spline2)
olsfit <- bols(x3)
attributes(olsfit)

form1 <- y ~ bbs(x1,knots=20,df=4) + bns(x2,knots=knots.x2,df=5)

# example for bspatial

x1 <- runif(250,-pi,pi)
x2 <- runif(250,-pi,pi)

y <- sin(x1)*sin(x2) + rnorm(250, sd = 0.4)

spline3 <- bspatial(x1, x2, xknots=12, yknots=12)
attributes(spline3)

form2 <- y ~ bspatial(x1, x2, xknots=12, yknots=12)

# decompose spatial effect into parametric part and deviation with 1 df

form2 <- y ~ bols(x1) + bols(x2) + bols(x1*x2) + 
             bspatial(x1, x2, xknots=12, yknots=12, center = TRUE, df=1)

# random intercept

id <- factor(rep(1:10, each=5))
raneff <- brandom(id)
attributes(raneff)

# random slope

z <- runif(50)
raneff <- brandom(id, z=z)
attributes(raneff)