krscv computes exhaustive cross-validation directed search for
a regression spline estimate of a one (1) dimensional dependent
variable on an r-dimensional vector of continuous and
nominal/ordinal (factor/ordered)
predictors.
krscv(xz,
y,
degree.max = 10,
segments.max = 10,
degree.min = 0,
segments.min = 1,
restarts = 0,
complexity = c("degree-knots","degree","knots"),
knots = c("quantiles","uniform","auto"),
basis = c("additive","tensor","glp","auto"),
cv.func = c("cv.ls","cv.gcv","cv.aic"),
degree = degree,
segments = segments,
tau = NULL,
weights = NULL,
singular.ok = FALSE)krscv returns a crscv object. Furthermore, the
function summary supports objects of this type. The
returned objects have the following components:
scalar/vector containing optimal degree(s) of spline or number of segments
vector/matrix of values of K evaluated during search
number of restarts during search, if any
optimal bandwidths for categorical predictors
vector/matrix of optimal bandwidths for each degree of spline
objective function value at optimum
vector of objective function values at each degree
of spline or number of segments in K.mat
continuous univariate vector
continuous and/or nominal/ordinal
(factor/ordered) predictors
the maximum degree of the B-spline basis for
each of the continuous predictors (default degree.max=10)
the maximum segments of the B-spline basis for
each of the continuous predictors (default segments.max=10)
the minimum degree of the B-spline basis for
each of the continuous predictors (default degree.min=0)
the minimum segments of the B-spline basis for
each of the continuous predictors (default segments.min=1)
number of times to restart optim from different initial
random values (default restarts=0) when searching for optimal
bandwidths for the categorical predictors for each unique K
combination (i.e.\ degree/segments)
a character string (default
complexity="degree-knots") indicating whether model
‘complexity’ is determined by the degree of the spline or by
the number of segments (‘knots’). This option allows the user
to use cross-validation to select either the spline degree (number
of knots held fixed) or the number of knots (spline degree held
fixed) or both the spline degree and number of knots
a character string (default knots="quantiles")
specifying where knots are to be placed. ‘quantiles’ specifies
knots placed at equally spaced quantiles (equal number of observations
lie in each segment) and ‘uniform’ specifies knots placed at
equally spaced intervals. If knots="auto", the knot type will
be automatically determined by cross-validation
a character string (default basis="additive")
indicating whether the additive or tensor product B-spline basis
matrix for a multivariate polynomial spline or generalized B-spline
polynomial basis should be used. Note this can be automatically
determined by cross-validation if cv=TRUE and
basis="auto", and is an ‘all or none’ proposition
(i.e. interaction terms for all predictors or for no predictors
given the nature of ‘tensor products’). Note also that if
there is only one predictor this defaults to basis="additive"
to avoid unnecessary computation as the spline bases are equivalent
in this case
a character string (default cv.func="cv.ls")
indicating which method to use to select smoothing
parameters. cv.gcv specifies generalized cross-validation
(Craven and Wahba (1979)), cv.aic specifies expected
Kullback-Leibler cross-validation (Hurvich, Simonoff, and Tsai
(1998)), and cv.ls specifies least-squares
cross-validation
integer/vector specifying the degree of the B-spline
basis for each dimension of the continuous x
integer/vector specifying the number of segments of
the B-spline basis for each dimension of the continuous x
(i.e. number of knots minus one)
if non-null a number in (0,1) denoting the quantile for which a quantile
regression spline is to be estimated rather than estimating the
conditional mean (default tau=NULL)
an optional vector of weights to be used in the fitting process. Should be ‘NULL’ or a numeric vector. If non-NULL, weighted least squares is used with weights ‘weights’ (that is, minimizing ‘sum(w*e^2)’); otherwise ordinary least squares is used.
a logical value (default singular.ok=FALSE) that, when
FALSE, discards singular bases during cross-validation (a check
for ill-conditioned bases is performed).
Jeffrey S. Racine racinej@mcmaster.ca
krscv computes exhaustive cross-validation for a regression
spline estimate of a one (1) dimensional dependent variable on an
r-dimensional vector of continuous and nominal/ordinal
(factor/ordered) predictors. The optimal
K/lambda combination is returned along with other
results (see below for return values). The method uses kernel
functions appropriate for categorical (ordinal/nominal) predictors
which avoids the loss in efficiency associated with sample-splitting
procedures that are typically used when faced with a mix of continuous
and nominal/ordinal (factor/ordered)
predictors.
For the continuous predictors the regression spline model employs
either the additive or tensor product B-spline basis matrix for a
multivariate polynomial spline via the B-spline routines in the GNU
Scientific Library (https://www.gnu.org/software/gsl/) and the
tensor.prod.model.matrix function.
For the discrete predictors the product kernel function is of the ‘Li-Racine’ type (see Li and Racine (2007) for details).
For each unique combination of degree and segment,
numerical search for the bandwidth vector lambda is undertaken
using optim and the box-constrained L-BFGS-B
method (see optim for details). The user may restart the
optim algorithm as many times as desired via the
restarts argument. The approach ascends from K=0 through
degree.max/segments.max and for each value of K
searches for the optimal bandwidths for this value of K. After
the most complex model has been searched then the optimal
K/lambda combination is selected. If any element of the
optimal K vector coincides with
degree.max/segments.max a warning is produced and the
user ought to restart their search with a larger value of
degree.max/segments.max.
Craven, P. and G. Wahba (1979), “Smoothing Noisy Data With Spline Functions,” Numerische Mathematik, 13, 377-403.
Hurvich, C.M. and J.S. Simonoff and C.L. Tsai (1998), “Smoothing Parameter Selection in Nonparametric Regression Using an Improved Akaike Information Criterion,” Journal of the Royal Statistical Society B, 60, 271-293.
Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.
Ma, S. and J.S. Racine and L. Yang (2015), “Spline Regression in the Presence of Categorical Predictors,” Journal of Applied Econometrics, Volume 30, 705-717.
Ma, S. and J.S. Racine (2013), “Additive Regression Splines with Irrelevant Categorical and Continuous Regressors,” Statistica Sinica, Volume 23, 515-541.
loess, npregbw,
set.seed(42)
## Simulated data
n <- 1000
x <- runif(n)
z <- round(runif(n,min=-0.5,max=1.5))
z.unique <- uniquecombs(as.matrix(z))
ind <- attr(z.unique,"index")
ind.vals <- sort(unique(ind))
dgp <- numeric(length=n)
for(i in 1:nrow(z.unique)) {
zz <- ind == ind.vals[i]
dgp[zz] <- z[zz]+cos(2*pi*x[zz])
}
y <- dgp + rnorm(n,sd=.1)
xdata <- data.frame(x,z=factor(z))
## Compute the optimal K and lambda, determine optimal number of knots, set
## spline degree for x to 3
cv <- krscv(x=xdata,y=y,complexity="knots",degree=c(3))
summary(cv)
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