set.seed(290875)
n <- 100
x1 <- rnorm(n)
x2 <- rnorm(n) + 0.25 * x1
x3 <- as.factor(sample(0:1, 100, replace = TRUE))
x4 <- gl(4, 25)
y <- 3 * sin(x1) + x2^2 + rnorm(n)
weights <- drop(rmultinom(1, n, rep.int(1, n) / n))
### set up base-learners
spline1 <- bbs(x1, knots = 20, df = 4)
extract(spline1, "design")[1:10, 1:10]
extract(spline1, "penalty")
knots.x2 <- quantile(x2, c(0.25, 0.5, 0.75))
spline2 <- bbs(x2, knots = knots.x2, df = 5)
ols3 <- bols(x3)
extract(ols3)
ols4 <- bols(x4)
### compute base-models
drop(ols3$dpp(weights)$fit(y)$model) ## same as:
coef(lm(y ~ x3, weights = weights))
drop(ols4$dpp(weights)$fit(y)$model) ## same as:
coef(lm(y ~ x4, weights = weights))
### fit model, component-wise
mod1 <- mboost_fit(list(spline1, spline2, ols3, ols4), y, weights)
### more convenient formula interface
mod2 <- mboost(y ~ bbs(x1, knots = 20, df = 4) +
bbs(x2, knots = knots.x2, df = 5) +
bols(x3) + bols(x4), weights = weights)
all.equal(coef(mod1), coef(mod2))
### grouped linear effects
# center x1 and x2 first
x1 <- scale(x1, center = TRUE, scale = FALSE)
x2 <- scale(x2, center = TRUE, scale = FALSE)
model <- gamboost(y ~ bols(x1, x2, intercept = FALSE) +
bols(x1, intercept = FALSE) +
bols(x2, intercept = FALSE),
control = boost_control(mstop = 50))
coef(model, which = 1) # one base-learner for x1 and x2
coef(model, which = 2:3) # two separate base-learners for x1 and x2
# zero because they were (not yet) selected.
### 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, knots = 12)
Xmat <- extract(spline3, "design")
## 12 inner knots + 4 boundary knots = 16 knots per direction
## THUS: 16 * 16 = 256 columns
dim(Xmat)
extract(spline3, "penalty")[1:10, 1:10]
## specify number of knots separately
form1 <- y ~ bspatial(x1, x2, knots = list(x1 = 12, x2 = 14))
## decompose spatial effect into parametric part and
## deviation with one df
form2 <- y ~ bols(x1) + bols(x2) + bols(x1, by = x2, intercept = FALSE) +
bspatial(x1, x2, knots = 12, center = TRUE, df = 1)
mod1 <- gamboost(form1)
plot(mod1)
mod2 <- gamboost(form2)
## automated plot function:
plot(mod2)
## plot sum of linear and smooth effects:
library(lattice)
df <- expand.grid(x1 = unique(x1), x2 = unique(x2))
df$pred <- predict(mod2, newdata = df)
levelplot(pred ~ x1 * x2, data = df)
## specify radial basis function base-learner for spatial effect
## and use data-adaptive effective range (theta = NULL, see 'args')
form3 <- y ~ brad(x1, x2)
## Now use different settings, e.g. 50 knots and theta fixed to 0.4
## (not really a good setting)
form4 <- y ~ brad(x1, x2, knots = 50, args = list(theta = 0.4))
mod3 <- gamboost(form3)
plot(mod3)
dim(extract(mod3, what = "design", which = "brad")[[1]])
knots <- attr(extract(mod3, what = "design", which = "brad")[[1]], "knots")
mod4 <- gamboost(form4)
dim(extract(mod4, what = "design", which = "brad")[[1]])
plot(mod4)
### random intercept
id <- factor(rep(1:10, each = 5))
raneff <- brandom(id)
extract(raneff, "design")
extract(raneff, "penalty")
## random intercept with non-observed category
set.seed(1907)
y <- rnorm(50, mean = rep(rnorm(10), each = 5), sd = 0.1)
plot(y ~ id)
# category 10 not observed
obs <- c(rep(1, 45), rep(0, 5))
model <- gamboost(y ~ brandom(id), weights = obs)
coef(model)
fitted(model)[46:50] # just the grand mean as usual for
# random effects models
### random slope
z <- runif(50)
raneff <- brandom(id, by = z)
extract(raneff, "design")
extract(raneff, "penalty")
### specify simple interaction model (with main effect)
n <- 210
x <- rnorm(n)
X <- model.matrix(~ x)
z <- gl(3, n/3)
Z <- model.matrix(~z)
beta <- list(c(0,1), c(-3,4), c(2, -4))
y <- rnorm(length(x), mean = (X * Z[,1]) %*% beta[[1]] +
(X * Z[,2]) %*% beta[[2]] +
(X * Z[,3]) %*% beta[[3]])
plot(y ~ x, col = z)
## specify main effect and interaction
mod_glm <- gamboost(y ~ bols(x) + bols(x, by = z),
control = boost_control(mstop = 100))
nd <- data.frame(x, z)
nd <- nd[order(x),]
nd$pred_glm <- predict(mod_glm, newdata = nd)
for (i in seq(along = levels(z)))
with(nd[nd$z == i,], lines(x, pred_glm, col = z))
mod_gam <- gamboost(y ~ bbs(x) + bbs(x, by = z, df = 8),
control = boost_control(mstop = 100))
nd$pred_gam <- predict(mod_gam, newdata = nd)
for (i in seq(along = levels(z)))
with(nd[nd$z == i,], lines(x, pred_gam, col = z, lty = "dashed"))
### convenience function for plotting
par(mfrow = c(1,3))
plot(mod_gam)
### remove intercept from base-learner
### and add explicit intercept to the model
tmpdata <- data.frame(x = 1:100, y = rnorm(1:100), int = rep(1, 100))
mod <- gamboost(y ~ bols(int, intercept = FALSE) +
bols(x, intercept = FALSE),
data = tmpdata,
control = boost_control(mstop = 1000))
cf <- unlist(coef(mod))
## add offset
cf[1] <- cf[1] + mod$offset
signif(cf, 3)
signif(coef(lm(y ~ x, data = tmpdata)), 3)
### much quicker and better with (mean-) centering
tmpdata$x_center <- tmpdata$x - mean(tmpdata$x)
mod_center <- gamboost(y ~ bols(int, intercept = FALSE) +
bols(x_center, intercept = FALSE),
data = tmpdata,
control = boost_control(mstop = 100))
cf_center <- unlist(coef(mod_center, which=1:2))
## due to the shift in x direction we need to subtract
## beta_1 * mean(x) to get the correct intercept
cf_center[1] <- cf_center[1] + mod_center$offset -
cf_center[2] * mean(tmpdata$x)
signif(cf_center, 3)
signif(coef(lm(y ~ x, data = tmpdata)), 3)
### large data set with ties
nunique <- 100
xindex <- sample(1:nunique, 1000000, replace = TRUE)
x <- runif(nunique)
y <- rnorm(length(xindex))
w <- rep.int(1, length(xindex))
### brute force computations
op <- options()
options(mboost_indexmin = Inf, mboost_useMatrix = FALSE)
## data pre-processing
b1 <- bbs(x[xindex])$dpp(w)
## model fitting
c1 <- b1$fit(y)$model
options(op)
### automatic search for ties, faster
b2 <- bbs(x[xindex])$dpp(w)
c2 <- b2$fit(y)$model
### manual specification of ties, even faster
b3 <- bbs(x, index = xindex)$dpp(w)
c3 <- b3$fit(y)$model
all.equal(c1, c2)
all.equal(c1, c3)
### cyclic P-splines
set.seed(781)
x <- runif(200, 0,(2*pi))
y <- rnorm(200, mean=sin(x), sd=0.2)
newX <- seq(0,2*pi, length=100)
### model without cyclic constraints
mod <- gamboost(y ~ bbs(x, knots = 20))
### model with cyclic constraints
mod_cyclic <- gamboost(y ~ bbs(x, cyclic=TRUE, knots = 20,
boundary.knots=c(0, 2*pi)))
par(mfrow = c(1,2))
plot(x,y, main="bbs (non-cyclic)", cex=0.5)
lines(newX, sin(newX), lty="dotted")
lines(newX + 2 * pi, sin(newX), lty="dashed")
lines(newX, predict(mod, data.frame(x = newX)),
col="red", lwd = 1.5)
lines(newX + 2 * pi, predict(mod, data.frame(x = newX)),
col="blue", lwd=1.5)
plot(x,y, main="bbs (cyclic)", cex=0.5)
lines(newX, sin(newX), lty="dotted")
lines(newX + 2 * pi, sin(newX), lty="dashed")
lines(newX, predict(mod_cyclic, data.frame(x = newX)),
col="red", lwd = 1.5)
lines(newX + 2 * pi, predict(mod_cyclic, data.frame(x = newX)),
col="blue", lwd = 1.5)
### use buser() to mimic p-spline base-learner:
set.seed(1907)
x <- rnorm(100)
y <- rnorm(100, mean = x^2, sd = 0.1)
mod1 <- gamboost(y ~ bbs(x))
## now extract design and penalty matrix
X <- extract(bbs(x), "design")
K <- extract(bbs(x), "penalty")
## use X and K in buser()
mod2 <- gamboost(y ~ buser(X, K))
max(abs(predict(mod1) - predict(mod2))) # same results
### use buser() to mimic penalized ordinal base-learner:
z <- as.ordered(sample(1:3, 100, replace=TRUE))
y <- rnorm(100, mean = as.numeric(z), sd = 0.1)
X <- extract(bols(z))
K <- extract(bols(z), "penalty")
index <- extract(bols(z), "index")
mod1 <- gamboost(y ~ buser(X, K, df = 1, index = index))
mod2 <- gamboost(y ~ bols(z, df = 1))
max(abs(predict(mod1) - predict(mod2))) # same results
### kronecker product for matrix-valued responses
data("volcano", package = "datasets")
layout(matrix(1:2, ncol = 2))
## estimate mean of image treating image as matrix
image(volcano, main = "data")
x1 <- 1:nrow(volcano)
x2 <- 1:ncol(volcano)
vol <- as.vector(volcano)
mod <- mboost(vol ~ bbs(x1, df = 3, knots = 10)%O%
bbs(x2, df = 3, knots = 10),
control = boost_control(nu = 0.25))
mod[250]
volf <- matrix(fitted(mod), nrow = nrow(volcano))
image(volf, main = "fitted")
## the old-fashioned way, a waste of space and time
x <- expand.grid(x1, x2)
modx <- mboost(vol ~ bbs(Var2, df = 3, knots = 10) %X%
bbs(Var1, df = 3, knots = 10), data = x,
control = boost_control(nu = 0.25))
modx[250]
max(abs(fitted(mod) - fitted(modx)))
### setting contrasts via contrasts.arg
x <- as.factor(sample(1:4, 100, replace = TRUE))
## compute base-learners with different reference categories
BL1 <- bols(x, contrasts.arg = contr.treatment(4, base = 1)) # default
BL2 <- bols(x, contrasts.arg = contr.treatment(4, base = 2))
## compute 'sum to zero contrasts' using character string
BL3 <- bols(x, contrasts.arg = "contr.sum")
## extract model matrices to check if it works
extract(BL1)
extract(BL2)
extract(BL3)
### setting contrasts using named lists in contrasts.arg
x2 <- as.factor(sample(1:4, 100, replace = TRUE))
BL4 <- bols(x, x2,
contrasts.arg = list(x = contr.treatment(4, base = 2),
x2 = "contr.helmert"))
extract(BL4)
### using special contrast: "contr.dummy":
BL5 <- bols(x, contrasts.arg = "contr.dummy")
extract(BL5)Run the code above in your browser using DataLab