npudensbw  R Documentation 
npudensbw
computes a bandwidth object for a pvariate
kernel unconditional density estimator defined over mixed continuous
and discrete (unordered, ordered) data using either the normal
reference ruleofthumb, likelihood crossvalidation, or leastsquares
cross validation using the method of Li and Racine (2003).
npudensbw(...) ## S3 method for class 'formula' npudensbw(formula, data, subset, na.action, call, ...) ## S3 method for class 'NULL' npudensbw(dat = stop("invoked without input data 'dat'"), bws, ...) ## S3 method for class 'bandwidth' npudensbw(dat = stop("invoked without input data 'dat'"), bws, bandwidth.compute = TRUE, nmulti, remin = TRUE, itmax = 10000, ftol = 1.490116e07, tol = 1.490116e04, small = 1.490116e05, lbc.dir = 0.5, dfc.dir = 3, cfac.dir = 2.5*(3.0sqrt(5)), initc.dir = 1.0, lbd.dir = 0.1, hbd.dir = 1, dfac.dir = 0.25*(3.0sqrt(5)), initd.dir = 1.0, lbc.init = 0.1, hbc.init = 2.0, cfac.init = 0.5, lbd.init = 0.1, hbd.init = 0.9, dfac.init = 0.375, scale.init.categorical.sample = FALSE, ...) ## Default S3 method: npudensbw(dat = stop("invoked without input data 'dat'"), bws, bandwidth.compute = TRUE, nmulti, remin, itmax, ftol, tol, small, lbc.dir, dfc.dir, cfac.dir, initc.dir, lbd.dir, hbd.dir, dfac.dir, initd.dir, lbc.init, hbc.init, cfac.init, lbd.init, hbd.init, dfac.init, scale.init.categorical.sample, bwmethod, bwscaling, bwtype, ckertype, ckerorder, ukertype, okertype, ...)
formula 
a symbolic description of variables on which bandwidth selection is to be performed. The details of constructing a formula are described below. 
data 
an optional data frame, list or environment (or object
coercible to a data frame by 
subset 
an optional vector specifying a subset of observations to be used in the fitting process. 
na.action 
a function which indicates what should happen when the data contain

call 
the original function call. This is passed internally by

dat 
a pvariate data frame on which bandwidth selection will be performed. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof. 
bws 
a bandwidth specification. This can be set as a bandwidth object
returned from a previous invocation, or as a vector of bandwidths,
with each element i corresponding to the bandwidth for column
i in 
... 
additional arguments supplied to specify the bandwidth type, kernel types, selection methods, and so on, detailed below. 
bwmethod 
a character string specifying the bandwidth selection
method. 
bwscaling 
a logical value that when set to 
bwtype 
character string used for the continuous variable bandwidth type,
specifying the type of bandwidth to compute and return in the

bandwidth.compute 
a logical value which specifies whether to do a numerical search for
bandwidths or not. If set to 
ckertype 
character string used to specify the continuous kernel type.
Can be set as 
ckerorder 
numeric value specifying kernel order (one of

ukertype 
character string used to specify the unordered categorical kernel type.
Can be set as 
okertype 
character string used to specify the ordered categorical kernel type.
Can be set as 
nmulti 
integer number of times to restart the process of finding extrema of the crossvalidation function from different (random) initial points. 
remin 
a logical value which when set as 
itmax 
integer number of iterations before failure in the numerical
optimization routine. Defaults to 
ftol 
fractional tolerance on the value of the crossvalidation function
evaluated at located minima (of order the machine precision or
perhaps slightly larger so as not to be diddled by
roundoff). Defaults to 
tol 
tolerance on the position of located minima of the crossvalidation
function (tol should generally be no smaller than the square root of
your machine's floating point precision). Defaults to 
small 
a small number used to bracket a minimum (it is hopeless to ask for
a bracketing interval of width less than sqrt(epsilon) times its
central value, a fractional width of only about 1004 (single
precision) or 3x108 (double precision)). Defaults to 
lbc.dir,dfc.dir,cfac.dir,initc.dir 
lower bound, chisquare
degrees of freedom, stretch factor, and initial nonrandom values
for direction set search for Powell's algorithm for 
lbd.dir,hbd.dir,dfac.dir,initd.dir 
lower bound, upper bound, stretch factor, and initial nonrandom values for direction set search for Powell's algorithm for categorical variables. See Details 
lbc.init, hbc.init, cfac.init 
lower bound, upper bound, and
nonrandom initial values for scale factors for 
lbd.init, hbd.init, dfac.init 
lower bound, upper bound, and nonrandom initial values for scale factors for categorical variables for Powell's algorithm. See Details 
scale.init.categorical.sample 
a logical value that when set
to 
Typical usages are (see below for a complete list of options and also the examples at the end of this help file)
Usage 1: compute a bandwidth object using the formula interface: bw < npudensbw(~y) Usage 2: compute a bandwidth object using the data frame interface and change the default kernel and order: fhat < npudensbw(tdat = y, ckertype="epanechnikov", ckerorder=4)
npudensbw
implements a variety of methods for choosing
bandwidths for multivariate (pvariate) distributions defined over
a set of possibly continuous and/or discrete (unordered, ordered)
data. The approach is based on Li and Racine (2003) who employ
‘generalized product kernels’ that admit a mix of continuous
and discrete data types.
The crossvalidation methods employ multivariate numerical search algorithms (direction set (Powell's) methods in multidimensions).
Bandwidths can (and will) differ for each variable which is, of course, desirable.
Three classes of kernel estimators for the continuous data types are available: fixed, adaptive nearestneighbor, and generalized nearestneighbor. Adaptive nearestneighbor bandwidths change with each sample realization in the set, x[i], when estimating the density at the point x. Generalized nearestneighbor bandwidths change with the point at which the density is estimated, x. Fixed bandwidths are constant over the support of x.
npudensbw
may be invoked either with a formulalike
symbolic description of variables on which bandwidth selection is to
be performed or through a simpler interface whereby data is
passed directly to the function via the dat
parameter. Use of
these two interfaces is mutually exclusive.
Data contained in the data frame dat
may be a mix of continuous
(default), unordered discrete (to be specified in the data frame
dat
using factor
), and ordered discrete (to be
specified in the data frame dat
using
ordered
). Data can be entered in an arbitrary order and
data types will be detected automatically by the routine (see
np
for details).
Data for which bandwidths are to be estimated may be specified
symbolically. A typical description has the form ~ data
, where
data
is a series of variables specified by name, separated by
the separation character '+'. For example, ~ x + y
specifies
that the bandwidths for the joint distribution of variables x
and y
are to be estimated. See below for further examples.
A variety of kernels may be specified by the user. Kernels implemented for continuous data types include the second, fourth, sixth, and eighth order Gaussian and Epanechnikov kernels, and the uniform kernel. Unordered discrete data types use a variation on Aitchison and Aitken's (1976) kernel, while ordered data types use a variation of the Wang and van Ryzin (1981) kernel.
The optimizer invoked for search is Powell's conjugate direction
method which requires the setting of (nonrandom) initial values and
search directions for bandwidths, and, when restarting, random values
for successive invocations. Bandwidths for numeric
variables
are scaled by robust measures of spread, the sample size, and the
number of numeric
variables where appropriate. Two sets of
parameters for bandwidths for numeric
can be modified, those
for initial values for the parameters themselves, and those for the
directions taken (Powell's algorithm does not involve explicit
computation of the function's gradient). The default values are set by
considering search performance for a variety of difficult test cases
and simulated cases. We highly recommend restarting search a large
number of times to avoid the presence of local minima (achieved by
modifying nmulti
). Further refinement for difficult cases can
be achieved by modifying these sets of parameters. However, these
parameters are intended more for the authors of the package to enable
‘tuning’ for various methods rather than for the user themselves.
npudensbw
returns a bandwidth
object, with the
following components:
bw 
bandwidth(s), scale factor(s) or nearest neighbours for the
data, 
fval 
objective function value at minimum 
if bwtype
is set to fixed
, an object containing
bandwidths, of class bandwidth
(or scale factors if bwscaling = TRUE
) is returned. If it is set to
generalized_nn
or adaptive_nn
, then instead the
kth nearest
neighbors are returned for the continuous variables while the discrete
kernel bandwidths are returned for the discrete variables. Bandwidths
are stored under the component name bw
, with each
element i corresponding to column i of input data
dat
.
The functions predict
, summary
and plot
support
objects of type bandwidth
.
If you are using data of mixed types, then it is advisable to use the
data.frame
function to construct your input data and not
cbind
, since cbind
will typically not work as
intended on mixed data types and will coerce the data to the same
type.
Caution: multivariate datadriven bandwidth selection methods are, by
their nature, computationally intensive. Virtually all methods
require dropping the ith observation from the data set, computing an
object, repeating this for all observations in the sample, then
averaging each of these leaveoneout estimates for a given
value of the bandwidth vector, and only then repeating this a large
number of times in order to conduct multivariate numerical
minimization/maximization. Furthermore, due to the potential for local
minima/maxima, restarting this procedure a large number of times may
often be necessary. This can be frustrating for users possessing
large datasets. For exploratory purposes, you may wish to override the
default search tolerances, say, setting ftol=.01 and tol=.01 and
conduct multistarting (the default is to restart min(5, ncol(dat))
times) as is done for a number of examples. Once the procedure
terminates, you can restart search with default tolerances using those
bandwidths obtained from the less rigorous search (i.e., set
bws=bw
on subsequent calls to this routine where bw
is
the initial bandwidth object). A version of this package using the
Rmpi
wrapper is under development that allows one to deploy
this software in a clustered computing environment to facilitate
computation involving large datasets.
Tristen Hayfield tristen.hayfield@gmail.com, Jeffrey S. Racine racinej@mcmaster.ca
Aitchison, J. and , C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method,” Biometrika, 63, 413420.
Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.
Li, Q. and J.S. Racine (2003), “Nonparametric estimation of distributions with categorical and continuous data,” Journal of Multivariate Analysis, 86, 266292.
Ouyang, D. and Q. Li and J.S. Racine (2006), “Crossvalidation and the estimation of probability distributions with categorical data,” Journal of Nonparametric Statistics, 18, 69100.
Pagan, A. and A. Ullah (1999), Nonparametric Econometrics, Cambridge University Press.
Scott, D.W. (1992), Multivariate Density Estimation. Theory, Practice and Visualization, New York: Wiley.
Silverman, B.W. (1986), Density Estimation, London: Chapman and Hall.
Wang, M.C. and J. van Ryzin (1981), “A class of smooth estimators for discrete distributions,” Biometrika, 68, 301309.
bw.nrd
, bw.SJ
, hist
,
npudens
, npudist
## Not run: # EXAMPLE 1 (INTERFACE=FORMULA): For this example, we load Giovanni # Baiocchi's Italian GDP panel (see Italy for details), then create a # data frame in which year is an ordered factor, GDP is continuous. data("Italy") attach(Italy) data < data.frame(ordered(year), gdp) # We compute bandwidths for the kernel density estimator using the # normalreference ruleofthumb. Otherwise, we use the defaults (second # order Gaussian kernel, fixed bandwidths). Note that the bandwidth # object you compute inherits all properties of the estimator (kernel # type, kernel order, estimation method) and can be fed directly into # the plotting utility plot() or into the npudens() function. bw < npudensbw(formula=~ordered(year)+gdp, bwmethod="normalreference") summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # Next, specify a value for the bandwidths manually (0.5 for the first # variable, 1.0 for the second)... bw < npudensbw(formula=~ordered(year)+gdp, bws=c(0.5, 1.0), bandwidth.compute=FALSE) summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # Next, if you wanted to use the 1.06 sigma n^{1/(2p+q)} ruleofthumb # for the bandwidth for the continuous variable and, say, no smoothing # for the discrete variable, you would use the bwscaling=TRUE argument # and feed in the values 0 for the first variable (year) and 1.06 for # the second (gdp). Note that in the printout it reports the `scale # factors' rather than the `bandwidth' as reported in some of the # previous examples. bw < npudensbw(formula=~ordered(year)+gdp, bws=c(0, 1.06), bwscaling=TRUE, bandwidth.compute=FALSE) summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # If you wished to use, say, an eighth order Epanechnikov kernel for the # continuous variables and specify your own bandwidths, you could do # that as follows. bw < npudensbw(formula=~ordered(year)+gdp, bws=c(0.5, 1.0), bandwidth.compute=FALSE, ckertype="epanechnikov", ckerorder=8) summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # If you preferred, say, nearestneighbor bandwidths and a generalized # kernel estimator for the continuous variable, you would use the # bwtype="generalized_nn" argument. bw < npudensbw(formula=~ordered(year)+gdp, bwtype = "generalized_nn") summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # Next, compute bandwidths using likelihood crossvalidation, fixed # bandwidths, and a second order Gaussian kernel for the continuous # variable (default). Note  this may take a few minutes depending on # the speed of your computer. bw < npudensbw(formula=~ordered(year)+gdp) summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # Finally, if you wish to use initial values for numerical search, you # can either provide a vector of bandwidths as in bws=c(...) or a # bandwidth object from a previous run, as in bw < npudensbw(formula=~ordered(year)+gdp, bws=c(1, 1)) summary(bw) detach(Italy) # EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we load Giovanni # Baiocchi's Italian GDP panel (see Italy for details), then create a # data frame in which year is an ordered factor, GDP is continuous. data("Italy") attach(Italy) data < data.frame(ordered(year), gdp) # We compute bandwidths for the kernel density estimator using the # normalreference ruleofthumb. Otherwise, we use the defaults (second # order Gaussian kernel, fixed bandwidths). Note that the bandwidth # object you compute inherits all properties of the estimator (kernel # type, kernel order, estimation method) and can be fed directly into # the plotting utility plot() or into the npudens() function. bw < npudensbw(dat=data, bwmethod="normalreference") summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # Next, specify a value for the bandwidths manually (0.5 for the first # variable, 1.0 for the second)... bw < npudensbw(dat=data, bws=c(0.5, 1.0), bandwidth.compute=FALSE) summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # Next, if you wanted to use the 1.06 sigma n^{1/(2p+q)} ruleofthumb # for the bandwidth for the continuous variable and, say, no smoothing # for the discrete variable, you would use the bwscaling=TRUE argument # and feed in the values 0 for the first variable (year) and 1.06 for # the second (gdp). Note that in the printout it reports the `scale # factors' rather than the `bandwidth' as reported in some of the # previous examples. bw < npudensbw(dat=data, bws=c(0, 1.06), bwscaling=TRUE, bandwidth.compute=FALSE) summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # If you wished to use, say, an eighth order Epanechnikov kernel for the # continuous variables and specify your own bandwidths, you could do # that as follows: bw < npudensbw(dat=data, bws=c(0.5, 1.0), bandwidth.compute=FALSE, ckertype="epanechnikov", ckerorder=8) summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # If you preferred, say, nearestneighbor bandwidths and a generalized # kernel estimator for the continuous variable, you would use the # bwtype="generalized_nn" argument. bw < npudensbw(dat=data, bwtype = "generalized_nn") summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # Next, compute bandwidths using likelihood crossvalidation, fixed # bandwidths, and a second order Gaussian kernel for the continuous # variable (default). Note  this may take a few minutes depending on # the speed of your computer. bw < npudensbw(dat=data) summary(bw) # Sleep for 5 seconds so that we can examine the output... Sys.sleep(5) # Finally, if you wish to use initial values for numerical search, you # can either provide a vector of bandwidths as in bws=c(...) or a # bandwidth object from a previous run, as in bw < npudensbw(dat=data, bws=c(1, 1)) summary(bw) detach(Italy) ## End(Not run)
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