Construct BayesX Model Terms in A Formula
Function sx is a model term constructor function for terms used within the formula
argument of function bayesx. The function does not evaluate matrices etc., the
behavior is similar to function s from package mgcv. It
purely exists to build a basic setup for the model term which can be processed by function
bayesx.construct.
sx(x, z = NULL, bs = "ps", by = NA, ...)
x |
the covariate the model term is a function of. |
z |
a second covariate. |
bs |
a |
by |
a |
... |
special controlling arguments or objects used for the model term, see also
the examples and function |
The following term types may be specified using argument bs:
"rw1", "rw2": Zero degree P-splines: Defines a zero degree P-spline with first or
second order difference penalty. A zero degree P-spline typically
estimates for every distinct covariate value in the dataset a separate
parameter. Usually there is no reason to prefer zero degree P-splines
over higher order P-splines. An exception are ordinal covariates or
continuous covariates with only a small number of different values.
For ordinal covariates higher order P-splines are not meaningful while
zero degree P-splines might be an alternative to modeling nonlinear
relationships via a dummy approach with completely unrestricted
regression parameters.
"season": Seasonal effect of a time scale.
"ps", "psplinerw1", "psplinerw2": P-spline with first or second order
difference penalty.
"te", "pspline2dimrw1": Defines a two-dimensional P-spline based on the tensor
product of one-dimensional P-splines with a two-dimensional first order random walk
penalty for the parameters of the spline.
"kr", "kriging": Kriging with stationary Gaussian random fields.
"gk", "geokriging": Geokriging with stationary Gaussian random fields: Estimation
is based on the centroids of a map object provided in
boundary format (see function read.bnd and shp2bnd) as an additional
argument named map within function sx, or supplied within argument
xt when using function s, e.g., xt = list(map = MapBnd).
"gs", "geospline": Geosplines based on two-dimensional P-splines with a
two-dimensional first order random walk penalty for the parameters of the spline.
Estimation is based on the coordinates of the centroids of the regions
of a map object provided in boundary format (see function read.bnd and
shp2bnd) as an additional argument named map (see above).
"mrf", "spatial": Markov random fields: Defines a Markov random field prior for a
spatial covariate, where geographical information is provided by a map object in
boundary or graph file format (see function read.bnd, read.gra and
shp2bnd), as an additional argument named map (see above).
"bl", "baseline": Nonlinear baseline effect in hazard regression or multi-state
models: Defines a P-spline with second order random walk penalty for the parameters of
the spline for the log-baseline effect log(λ(time)).
"factor": Special BayesX specifier for factors, especially meaningful if
method = "STEP", since the factor term is then treated as a full term,
which is either included or removed from the model.
"ridge", "lasso", "nigmix": Shrinkage of fixed effects: defines a
shrinkage-prior for the corresponding parameters
γ_j, j = 1, …, q, q ≥q 1 of the
linear effects x_1, …, x_q. There are three
priors possible: ridge-, lasso- and Normal Mixture
of inverse Gamma prior.
"re": Gaussian i.i.d. Random effects of a unit or cluster identification covariate.
A list of class "xx.smooth.spec", where "xx" is a basis/type identifying code
given by the bs argument of f.
Some care has to be taken with the identifiability of varying coefficients terms. The standard in
BayesX is to center nonlinear main effects terms around zero whereas varying coefficient terms are
not centered. This makes sense since main effects nonlinear terms are not identifiable and varying
coefficients terms are usually identifiable. However, there are situations where a varying
coefficients term is not identifiable. Then the term must be centered. Since centering is not
automatically accomplished it has to be enforced by the user by adding option
center = TRUE in function f. To give an example, the varying coefficient terms in
η = … + g_1(z_1)z + g_2(z_2)z + γ_0 + γ_1 z + … are not
identified, whereas in η = … + g_1(z_1)z + γ_0 + …, the varying
coefficient term is identifiable. In the first case, centering is necessary, in the second case,
it is not.
Nikolaus Umlauf, Thomas Kneib, Stefan Lang, Achim Zeileis.
## funktion sx() returns a list
## which is then processed by function
## bayesx.construct to build the
## BayesX model term structure
sx(x)
bayesx.construct(sx(x))
bayesx.construct(sx(x, bs = "rw1"))
bayesx.construct(sx(x, bs = "factor"))
bayesx.construct(sx(x, bs = "offset"))
bayesx.construct(sx(x, z, bs = "te"))
## varying coefficients
bayesx.construct(sx(x1, by = x2))
bayesx.construct(sx(x1, by = x2, center = TRUE))
## using a map for markov random fields
data("FantasyBnd")
plot(FantasyBnd)
bayesx.construct(sx(id, bs = "mrf", map = FantasyBnd))
## random effects
bayesx.construct(sx(id, bs = "re"))
## examples using optional controlling
## parameters and objects
bayesx.construct(sx(x, bs = "ps", knots = 20))
bayesx.construct(sx(x, bs = "ps", nrknots = 20))
bayesx.construct(sx(x, bs = "ps", knots = 20, nocenter = TRUE))
## use of bs with original
## BayesX syntax
bayesx.construct(sx(x, bs = "psplinerw1"))
bayesx.construct(sx(x, bs = "psplinerw2"))
bayesx.construct(sx(x, z, bs = "pspline2dimrw2"))
bayesx.construct(sx(id, bs = "spatial", map = FantasyBnd))
bayesx.construct(sx(x, z, bs = "kriging"))
bayesx.construct(sx(id, bs = "geospline", map = FantasyBnd, nrknots = 5))
bayesx.construct(sx(x, bs = "catspecific"))
## Not run:
## generate some data
set.seed(111)
n <- 200
## regressor
dat <- data.frame(x = runif(n, -3, 3))
## response
dat$y <- with(dat, 1.5 + sin(x) + rnorm(n, sd = 0.6))
## estimate models with
## bayesx REML and MCMC
b1 <- bayesx(y ~ sx(x), method = "REML", data = dat)
## increase inner knots
## decrease degree of the P-spline
b2 <- bayesx(y ~ sx(x, knots = 30, degree = 2), method = "REML", data = dat)
## compare reported output
summary(c(b1, b2))
## plot the effect for both models
plot(c(b1, b2), residuals = TRUE)
## more examples
set.seed(111)
n <- 500
## regressors
dat <- data.frame(x = runif(n, -3, 3), z = runif(n, -3, 3),
w = runif(n, 0, 6), fac = factor(rep(1:10, n/10)))
## response
dat$y <- with(dat, 1.5 + sin(x) + cos(z) * sin(w) +
c(2.67, 5, 6, 3, 4, 2, 6, 7, 9, 7.5)[fac] + rnorm(n, sd = 0.6))
## estimate model
b <- bayesx(y ~ sx(x) + sx(z, w, bs = "te") + fac,
data = dat, method = "MCMC")
summary(b)
plot(b)
## now a mrf example
## note: the regional identification
## covariate and the map regionnames
## should be coded as integer
set.seed(333)
## simulate some geographical data
data("MunichBnd")
N <- length(MunichBnd); n <- N*5
names(MunichBnd) <- 1:N
## regressors
dat <- data.frame(x1 = runif(n, -3, 3),
id = as.factor(rep(names(MunichBnd), length.out = n)))
dat$sp <- with(dat, sort(runif(N, -2, 2), decreasing = TRUE)[id])
## response
dat$y <- with(dat, 1.5 + sin(x1) + sp + rnorm(n, sd = 1.2))
## estimate models with
## bayesx MCMC and REML
b <- bayesx(y ~ sx(x1) + sx(id, bs = "mrf", map = MunichBnd),
method = "REML", data = dat)
## summary statistics
summary(b)
## plot the effects
op <- par(no.readonly = TRUE)
par(mfrow = c(1,2))
plot(b, term = "sx(id)", map = MunichBnd,
main = "bayesx() estimate")
plotmap(MunichBnd, x = dat$sp, id = dat$id,
main = "Truth")
par(op)
## model with random effects
set.seed(333)
N <- 30
n <- N*10
## regressors
dat <- data.frame(id = sort(rep(1:N, n/N)), x1 = runif(n, -3, 3))
dat$re <- with(dat, rnorm(N, sd = 0.6)[id])
## response
dat$y <- with(dat, 1.5 + sin(x1) + re + rnorm(n, sd = 0.6))
## estimate model
b <- bayesx(y ~ sx(x1, bs = "psplinerw1") + sx(id, bs = "re"), data = dat)
summary(b)
plot(b)
## extract estimated random effects
## and compare with true effects
plot(fitted(b, term = "sx(id)")$Mean ~ unique(dat$re))
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