clsd computes the logspline density, density
derivative, distribution, and smoothed quantiles for a one (1)
dimensional continuous variable using the approach of Racine
(2013).
clsd(x = NULL,
beta = NULL,
xeval = NULL,
degree = NULL,
segments = NULL,
degree.min = 2,
degree.max = 25,
segments.min = 1,
segments.max = 100,
lbound = NULL,
ubound = NULL,
basis = "tensor",
knots = "quantiles",
penalty = NULL,
deriv.index = 1,
deriv = 1,
elastic.max = TRUE,
elastic.diff = 3,
do.gradient = TRUE,
er = NULL,
monotone = TRUE,
monotone.lb = -250,
n.integrate = 500,
nmulti = 1,
method = c("L-BFGS-B", "Nelder-Mead", "BFGS", "CG", "SANN"),
verbose = FALSE,
quantile.seq = seq(.01,.99,by=.01),
random.seed = 42,
maxit = 10^5,
max.attempts = 25,
NOMAD = FALSE)a numeric vector of training data
a numeric vector of coefficients (default NULL)
a numeric vector of evaluation data
integer/vector specifying the polynomial degree of the
B-spline basis for each dimension of the continuous x (default
degree=2)
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) (default segments=1, i.e. Bezier
curve)
when elastic.max=FALSE, the
minimum/maximum segments of the B-spline basis for each of the
continuous predictors (default
segments.min=1,segments.max=100)
when elastic.max=FALSE the
minimum/maximum degree of the B-spline basis for each of the
continuous predictors (default degree.min=2,
degree.max=25)
lower/upper bound for the support of the density. For example, if there is a priori knowledge that the density equals zero to the left of 0, and has a discontinuity at 0, the user could specify lbound = 0. However, if the density is essentially zero near 0, one does not need to specify lbound
a character string (default basis="tensor")
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
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
an integer l (default deriv=1) specifying
whether to compute the univariate lth partial derivative for
each continuous predictor (and difference in levels for each
categorical predictor) or not and if so what order. Note that if
deriv is higher than the spline degree of the associated
continuous predictor then the derivative will be zero and a warning
issued to this effect
an integer l (default deriv.index=1)
specifying the index (currently only supports 1) of the variable whose
derivative is requested
integer number of times to restart the process of finding extrema of
the cross-validation function from different (random) initial
points (default nmulti=1)
the parameter to be used in the AIC criterion. The
method chooses the number of degrees plus number of segments
(knots-1) that maximizes 2*logl-penalty*(degree+segments). The
default is to use the penalty parameter of log(n)/2 (2
would deliver standard AIC, log(n) standard BIC)
a logical value/integer indicating
whether to use ‘elastic’ search bounds such that the optimal
degree/segment must lie elastic.diff units from the
respective search bounds
a logical value indicating whether or not to use
the analytical gradient during optimization (defaults to TRUE)
a scalar indicating the fraction of data range to extend
the tails (default 1/log(n), see extendrange for
further details)
a logical value indicating whether modify
the standard B-spline basis function so that it is tailored for
density estimation (default TRUE)
a negative bound specifying the lower bound on
the linear segment coefficients used when (monotone=FALSE)
the number of evenly spaced integration points on the extended range specified by er (defaults to 500)
see optim for details
a logical value which when TRUE produces verbose output
during optimization
a sequence of numbers lying in \([0,1]\) on which quantiles from the logspline distribution are obtained
seeds the random number generator for initial
parameter values when optim is called
maximum number of iterations used by optim
maximum number of attempts to undertake if optim
fails for any set of initial parameters for each value of
nmulti
a logical value which when TRUE calls snomadr
to determine the optimal degree and segments
clsd returns a clsd object. The generic functions
coef, fitted, plot and
summary support objects of this type (er=FALSE
plots the density on the sample realizations (default is ‘extended
range’ data), see er above, distribution=TRUE plots
the distribution). The returned object has the following components:
estimates of the density function at the sample points
the density evaluated on the ‘extended range’ of the data
estimates of the derivative of the density function at the sample points
estimates of the derivative of the density function evaluated on the ‘extended range’ of the data
estimates of the distribution function at the sample points
the distribution evaluated on the ‘extended range’ of the data
the ‘extended range’ of the data
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
vector of quantiles
vector generated by quantile.seq or input by the
user (lying in [0,1]) from which the quantiles xq are
obtained
This function should be considered to be in ‘beta’ status until further notice.
If smoother estimates are desired and degree=degree.min, increase
degree.min to, say, degree.min=3.
The use of ‘regression’ B-splines can lead to undesirable behavior at
the endpoints of the data (i.e. when monotone=FALSE). The
default ‘density’ B-splines ought to be well-behaved in these regions.
Typical usages are (see below for a list of options and also the examples at the end of this help file)
model <- clsd(x)
clsd computes a logspline density estimate of a one (1)
dimensional continuous variable.
The spline model employs the 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.
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.
Racine, J.S. (2013), “Logspline Mixed Data Density Estimation,” manuscript.
# NOT RUN {
## Old Faithful eruptions data histogram and clsd density
library(MASS)
data(faithful)
attach(faithful)
model <- clsd(eruptions)
ylim <- c(0,max(model$density,hist(eruptions,breaks=20,plot=FALSE)$density))
plot(model,ylim=ylim)
hist(eruptions,breaks=20,freq=FALSE,add=TRUE,lty=2)
rug(eruptions)
summary(model)
coef(model)
## Simulated data
set.seed(42)
require(logspline)
## Example - simulated data
n <- 250
x <- sort(rnorm(n))
f.dgp <- dnorm(x)
model <- clsd(x)
## Standard (cubic) estimate taken from the logspline package
## Compute MSEs
mse.clsd <- mean((fitted(model)-f.dgp)^2)
model.logspline <- logspline(x)
mse.logspline <- mean((dlogspline(x,model.logspline)-f.dgp)^2)
ylim <- c(0,max(fitted(model),dlogspline(x,model.logspline),f.dgp))
plot(model,
ylim=ylim,
sub=paste("MSE: logspline = ",format(mse.logspline),", clsd = ",
format(mse.clsd)),
lty=3,
col=3)
xer <- model$xer
lines(xer,dlogspline(xer,model.logspline),col=2,lty=2)
lines(xer,dnorm(xer),col=1,lty=1)
rug(x)
legend("topright",c("DGP",
paste("Cubic Logspline Density (package 'logspline', knots = ",
model.logspline$nknots,")",sep=""),
paste("clsd Density (degree = ", model$degree, ", segments = ",
model$segments,", penalty = ",round(model$penalty,2),")",sep="")),
lty=1:3,
col=1:3,
bty="n",
cex=0.75)
summary(model)
coef(model)
## Simulate data with known bounds
set.seed(42)
n <- 10000
x <- runif(n,0,1)
model <- clsd(x,lbound=0,ubound=1)
plot(model)
# }
# NOT RUN {
# }
# NOT RUN {
<!-- %% End dontrun -->
# }