Categorical Logspline Density
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)
x: a numeric vector of training data
beta: a numeric vector of coefficients (default NULL)
xeval: a numeric vector of evaluation data
degree: integer/vector specifying the polynomial degree of the B-spline basis for each dimension of the continuous x (default degree=2)
segments: 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)
segments.min,segments.max: 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)
degree.min,degree.max: 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)
lbound,ubound: 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
basis: 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
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
deriv: 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
deriv.index: an integer l (default deriv.index=1) specifying the index (currently only supports 1) of the variable whose derivative is requested
nmulti: integer number of times to restart the process of finding extrema of the cross-validation function from different (random) initial points (default nmulti=1)
penalty: 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)
elastic.max,elastic.diff: 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
do.gradient: a logical value indicating whether or not to use the analytical gradient during optimization (defaults to TRUE)
er: a scalar indicating the fraction of data range to extend the tails (default 1/log(n), see extendrange for further details)
monotone: a logical value indicating whether modify the standard B-spline basis function so that it is tailored for density estimation (default TRUE)
monotone.lb: a negative bound specifying the lower bound on the linear segment coefficients used when (monotone=FALSE)
n.integrate: the number of evenly spaced integration points on the extended range specified by er (defaults to 500)
method: see optim for details
verbose: a logical value which when TRUE produces verbose output during optimization
quantile.seq: a sequence of numbers lying in on which quantiles from the logspline distribution are obtained
random.seed: seeds the random number generator for initial parameter values when optim is called
maxit: maximum number of iterations used by optim
max.attempts: maximum number of attempts to undertake if optim
fails for any set of initial parameters for each value of nmulti
NOMAD: a logical value which when TRUE calls snomadr
to determine the optimal degree and segments
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.
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:
density: estimates of the density function at the sample points
density.er: the density evaluated on the extended range
of the data
density.deriv: estimates of the derivative of the density function at the sample points
density.deriv.er: estimates of the derivative of the density function evaluated on the extended range of the data
distribution: estimates of the distribution function at the sample points
distribution.er: the distribution evaluated on the extended range
of the data
xer: the extended range of the data
degree: integer/vector specifying the degree of the B-spline basis for each dimension of the continuous x
segments: integer/vector specifying the number of segments of the B-spline basis for each dimension of the continuous x
xq: vector of quantiles
tau: 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.
Racine, J.S. (2013), Logspline Mixed Data Density Estimation,
manuscript.
Jeffrey S. Racine racinej@mcmaster.ca
logspline
## 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) ## End(Not run)
Useful links