Categorical Regression Splines
Description
crs
computes a regression spline estimate of a
one (1) dimensional dependent variable on an r
dimensional
vector of continuous and categorical
(factor
/ordered
) predictors (Ma and
Racine (2013), Ma, Racine and Yang (under revision)).
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60  crs(...)
## Default S3 method:
crs(xz,
y,
degree = NULL,
segments = NULL,
include = NULL,
kernel = TRUE,
lambda = NULL,
complexity = c("degreeknots","degree","knots"),
knots = c("quantiles","uniform","auto"),
basis = c("additive","tensor","glp","auto"),
deriv = 0,
data.return = FALSE,
prune = FALSE,
model.return = FALSE,
tau = NULL,
weights = NULL,
...)
## S3 method for class 'formula'
crs(formula,
data = list(),
degree = NULL,
segments = NULL,
include = NULL,
degree.max = 10,
segments.max = 10,
degree.min = 0,
segments.min = 1,
cv.df.min = 1,
cv = c("nomad","exhaustive","none"),
cv.threshold = 1000,
cv.func = c("cv.ls","cv.gcv","cv.aic"),
kernel = TRUE,
lambda = NULL,
lambda.discrete = FALSE,
lambda.discrete.num = 100,
complexity = c("degreeknots","degree","knots"),
knots = c("quantiles","uniform","auto"),
basis = c("auto","additive","tensor","glp"),
deriv = 0,
data.return = FALSE,
prune = FALSE,
model.return = FALSE,
restarts = 0,
random.seed = 42,
max.bb.eval = 10000,
initial.mesh.size.real = "r1.0e01",
initial.mesh.size.integer = "1",
min.mesh.size.real = paste(sqrt(.Machine$double.eps)),
min.mesh.size.integer = paste(sqrt(.Machine$double.eps)),
min.poll.size.real = paste(sqrt(.Machine$double.eps)),
min.poll.size.integer = paste(sqrt(.Machine$double.eps)),
opts=list(),
nmulti = 5,
tau = NULL,
weights = NULL,
singular.ok = FALSE,
...)

Arguments
xz 
numeric ( 
y 
a numeric vector of responses. 
degree 
integer/vector specifying the polynomial degree of the
Bspline basis for each dimension of the continuous 
segments 
integer/vector specifying the number of segments of the
Bspline basis for each dimension of the continuous 
include 
integer/vector specifying whether each of the
nominal/ordinal ( 
lambda 
a vector of bandwidths for each dimension of the
categorical 
lambda.discrete 
if 
lambda.discrete.num 
a positive integer indicating the number of
discrete values that lambda can assume  this parameter will only be
used when 
formula 
a symbolic description of the model to be fit 
data 
an optional data frame containing the variables in the model 
cv 
a character string (default 
cv.threshold 
an integer (default 
cv.func 
a character string (default 
kernel 
a logical value (default 
degree.max 
the maximum degree of the Bspline basis for
each of the continuous predictors (default 
segments.max 
the maximum segments of the Bspline basis for
each of the continuous predictors (default 
degree.min 
the minimum degree of the Bspline basis for
each of the continuous predictors (default 
segments.min 
the minimum segments of the Bspline basis for
each of the continuous predictors (default 
cv.df.min 
the minimum degrees of freedom to allow when
conducting NOMADbased crossvalidation (default

complexity 
a character string (default
For the continuous predictors the regression spline model employs
either the additive or tensor product Bspline basis matrix for a
multivariate polynomial spline via the Bspline routines in the GNU
Scientific Library (http://www.gnu.org/software/gsl/) and the

knots 
a character string (default 
basis 
a character string (default 
deriv 
an integer 
data.return 
a logical value indicating whether to return

prune 
a logical value (default 
model.return 
a logical value indicating whether to return the
list of 
restarts 
integer specifying the number of times to restart the process of finding extrema of the crossvalidation function (for the bandwidths only) from different (random) initial points 
random.seed 
when it is not missing and not equal to 0, the
initial points will be generated using this seed when using

max.bb.eval 
argument passed to the NOMAD solver (see 
initial.mesh.size.real 
argument passed to the NOMAD solver (see 
initial.mesh.size.integer 
argument passed to the NOMAD solver (see 
min.mesh.size.real 
argument passed to the NOMAD solver (see 
min.mesh.size.integer 
arguments passed to the NOMAD solver (see 
min.poll.size.real 
arguments passed to the NOMAD solver (see 
min.poll.size.integer 
arguments passed to the NOMAD solver (see 
opts 
list of optional arguments to be passed to

nmulti 
integer number of times to restart the process of finding extrema of
the crossvalidation function from different (random) initial
points (default 
tau 
if nonnull 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 
weights 
an optional vector of weights to be used in the fitting process. Should be ‘NULL’ or a numeric vector. If nonNULL, weighted least squares is used with weights ‘weights’ (that is, minimizing ‘sum(w*e^2)’); otherwise ordinary least squares is used. 
singular.ok 
a logical value (default 
... 
optional arguments 
Details
Typical usages are (see below for a list of options and also the examples at the end of this help file)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  ## Estimate the model and let the basis type be determined by
## crossvalidation (i.e. crossvalidation will determine whether to
## use the additive, generalized, or tensor product basis)
model < crs(y~x1+x2)
## Estimate the model for a specified degree/segment/bandwidth
## combination and do not run crossvalidation (will use the
## additive basis by default)
model < crs(y~x1+factor(x2),cv="none",degree=3,segments=1,lambda=.1)
## Plot the mean and (asymptotic) error bounds
plot(model,mean=TRUE,ci=TRUE)
## Plot the first partial derivative and (asymptotic) error bounds
plot(model,deriv=1,ci=TRUE)

crs
computes a regression spline estimate of a one (1)
dimensional dependent variable on an r
dimensional vector of
continuous and categorical
(factor
/ordered
) predictors.
The regression spline model employs the tensor product Bspline basis
matrix for a multivariate polynomial spline via the Bspline routines
in the GNU Scientific Library (http://www.gnu.org/software/gsl/)
and the tensor.prod.model.matrix
function.
When basis="additive"
the model becomes additive in nature
(i.e. no interaction/tensor terms thus semiparametric not fully
nonparametric).
When basis="tensor"
the model uses the multivariate tensor
product basis.
When kernel=FALSE
the model uses indicator basis functions for
the nominal/ordinal (factor
/ordered
)
predictors rather than kernel weighting.
When kernel=TRUE
the product kernel function for the discrete
predictors is of the ‘LiRacine’ type (see Li and Racine (2007)
for details).
When cv="nomad"
, numerical search is undertaken using Nonsmooth
Optimization by Mesh Adaptive Direct Search (Abramson, Audet, Couture,
Dennis, Jr., and Le Digabel (2011)).
When kernel=TRUE
and cv="exhaustive"
, numerical search
is undertaken using optim
and the boxconstrained
LBFGSB
method (see optim
for details). The user
may restart the algorithm as many times as desired via the
restarts
argument (default restarts=0
). The approach
ascends from degree=0
(or segments=0
) through
degree.max
and for each value of degree
(or
segments
) searches for the optimal bandwidths. After the most
complex model has been searched then the optimal
degree
/segments
/lambda
combination is
selected. If any element of the optimal degree
(or
segments
) vector coincides with degree.max
(or segments.max
) a warning
is produced and the user ought to restart their search with a larger
value of degree.max
(or segments.max
).
Note that the default plot
method for a crs
object
provides some diagnostic measures, in particular, a) residuals versus
fitted values (useful for checking the assumption that
E(ux)=0
), b) a normal quantilequantile plot which allows
residuals to be assessed for normality (qqnorm
), c) a
scalelocation plot that is useful for checking the assumption that
the errors are iid and, in particular, that the variance is
homogeneous, and d) ‘Cook's distance’ which computes the
singlecase influence function. See below for other arguments for the
plot function for a crs
object.
Note that setting prune=TRUE
produces a final ‘pruning’
of the model via a stepwise crossvalidation criterion achieved by
modifying stepAIC
and replacing extractAIC
with extractCV
throughout the function. This option may be
enabled to remove potentially superfluous bases thereby improving the
finitesample efficiency of the resulting model. Note that if the
crossvalidation score for the pruned model is no better than that for
the original model then the original model is returned with a warning
to this effect. Note also that this option can only be used when
kernel=FALSE
.
Value
crs
returns a crs
object. The generic functions
fitted
and residuals
extract (or
generate) estimated values and residuals. Furthermore, the functions
summary
, predict
, and plot
(options mean=FALSE
, deriv=i
where i is an
integer, ci=FALSE
, persp.rgl=FALSE
,
plot.behavior=c("plot","plotdata","data")
,
xtrim=0.0
,xq=0.5
) support objects of this type. The
returned object has the following components:
fitted.values 
estimates of the regression function (conditional mean) at the sample points or evaluation points 
lwr,upr 
lower/upper bound for a 95% confidence interval for
the 
residuals 
residuals computed at the sample points or evaluation points 
degree 
integer/vector specifying the degree of the Bspline
basis for each dimension of the continuous 
segments 
integer/vector specifying the number of segments of
the Bspline basis for each dimension of the continuous 
include 
integer/vector specifying whether each of the
nominal/ordinal ( 
kernel 
a logical value indicating whether kernel smoothing was
used ( 
lambda 
vector of bandwidths used if 
call 
a symbolic description of the model 
r.squared 
coefficient of determination (Doksum and Samarov (1995)) 
model.lm 
an object of ‘ 
deriv.mat 
a matrix of derivatives (or differences in levels
for the categorical 
deriv.mat.lwr 
a matrix of 95% coverage lower bounds for

deriv.mat.upr 
a matrix of 95% coverage upper bounds for

hatvalues 
the 
P.hat 
the kernel probability estimates corresponding to the categorical predictors in the estimated model 
Usage Issues
Note that when kernel=FALSE
summary
supports the
option sigtest=TRUE
that conducts an Ftest for significance
for each predictor.
Author(s)
Jeffrey S. Racine racinej@mcmaster.ca
References
Abramson, M.A. and C. Audet and G. Couture and J.E. Dennis Jr. and and S. Le Digabel (2011), “The NOMAD project”. Software available at http://www.gerad.ca/nomad.
Craven, P. and G. Wahba (1979), “Smoothing Noisy Data With Spline Functions,” Numerische Mathematik, 13, 377403.
Doksum, K. and A. Samarov (1995), “Nonparametric Estimation of Global Functionals and a Measure of the Explanatory Power of Covariates in Regression,” The Annals of Statistics, 23 14431473.
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, 271293.
Le Digabel, S. (2011), “Algorithm 909: NOMAD: Nonlinear Optimization With The MADS Algorithm”. ACM Transactions on Mathematical Software, 37(4):44:144:15.
Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.
Ma, S. and J.S. Racine and L. Yang (under revision), “Spline Regression in the Presence of Categorical Predictors,” Journal of Applied Econometrics.
Ma, S. and J.S. Racine (2013), “Additive Regression Splines with Irrelevant Categorical and Continuous Regressors,” Statistica Sinica, Volume 23, 515541.
Racine, J.S. (2011), “CrossValidated Quantile Regression Splines,” manuscript.
See Also
smooth.spline
, loess
, npreg
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59  set.seed(42)
## Example  simulated data
n < 1000
num.eval < 50
x1 < runif(n)
x2 < runif(n)
z < rbinom(n,1,.5)
dgp < cos(2*pi*x1)+sin(2*pi*x2)+z
z < factor(z)
y < dgp + rnorm(n,sd=.5)
## Estimate a model with specified degree, segments, and bandwidth
model < crs(y~x1+x2+z,degree=c(5,5),
segments=c(1,1),
lambda=0.1,
cv="none",
kernel=TRUE)
summary(model)
## Perspective plot
x1.seq < seq(min(x1),max(x1),length=num.eval)
x2.seq < seq(min(x2),max(x2),length=num.eval)
x.grid < expand.grid(x1.seq,x2.seq)
newdata < data.frame(x1=x.grid[,1],x2=x.grid[,2],
z=factor(rep(0,num.eval**2),levels=c(0,1)))
z0 < matrix(predict(model,newdata=newdata),num.eval,num.eval)
newdata < data.frame(x1=x.grid[,1],x2=x.grid[,2],
z=factor(rep(1,num.eval),levels=c(0,1)))
z1 < matrix(predict(model,newdata=newdata),num.eval,num.eval)
zlim=c(min(z0,z1),max(z0,z1))
persp(x=x1.seq,y=x2.seq,z=z0,
xlab="x1",ylab="x2",zlab="y",zlim=zlim,
ticktype="detailed",
border="red",
theta=45,phi=45)
par(new=TRUE)
persp(x=x1.seq,y=x2.seq,z=z1,
xlab="x1",ylab="x2",zlab="y",zlim=zlim,
theta=45,phi=45,
ticktype="detailed",
border="blue")
## Partial regression surface plot
plot(model,mean=TRUE,ci=TRUE)
## Not run:
## A plot example where we extract the partial surfaces, confidence
## intervals etc. automatically generated by plot(mean=TRUE,...) but do
## not plot, rather save for separate use.
pdat < plot(model,mean=TRUE,ci=TRUE,plot.behavior="data")
## Column 1 is the (evaluation) predictor ([,1]), 24 ([,1]) the mean,
## lwr, and upr (note the returned value is a `list' hence pdat[[1]] is
## data for the first predictor, pdat[[2]] the second etc). Note that
## matplot() can plot this nicely.
matplot(pdat[[1]][,1],pdat[[1]][,1],
xlab=names(pdat[[1]][1]),ylab=names(pdat[[1]][2]),
lty=c(1,2,2),col=c(1,2,2),type="l")
## End(Not run)

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