Kernel Density Bandwidth Selection with Mixed Data Types

Description

npudensbw computes a bandwidth object for a p-variate kernel unconditional density estimator defined over mixed continuous and discrete (unordered, ordered) data using either the normal reference rule-of-thumb, likelihood cross-validation, or least-squares cross validation using the method of Li and Racine (2003).

Usage

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=1.19209e-07,
          tol=1.49012e-08,
          small=2.22045e-16,
          ...)

## Default S3 method:
npudensbw(dat = stop("invoked without input data 'dat'"),
          bws,
          bandwidth.compute = TRUE,
          nmulti,
          remin,
          itmax,
          ftol,
          tol,
          small,
          bwmethod,
          bwscaling,
          bwtype,
          ckertype,
          ckerorder,
          ukertype,
          okertype,
          ...) 

Arguments

Details

npudensbw implements a variety of methods for choosing bandwidths for multivariate (p-variate) 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 cross-validation 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 nearest-neighbor, and generalized nearest-neighbor. Adaptive nearest-neighbor bandwidths change with each sample realization in the set, x[i], when estimating the density at the point x. Generalized nearest-neighbor 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 formula-like 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.

Value

npudensbw returns a bandwidth object, with the following components:

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.

Usage Issues

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 data-driven 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 leave-one-out 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.

Author(s)

Tristen Hayfield hayfield@phys.ethz.ch, Jeffrey S. Racine racinej@mcmaster.ca

References

Aitchison, J. and , C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method,” Biometrika, 63, 413-420.

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, 266-292.

Ouyang, D. and Q. Li and J.S. Racine (2006), “Cross-validation and the estimation of probability distributions with categorical data,” Journal of Nonparametric Statistics, 18, 69-100.

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, 301-309.

See Also

bw.nrd, bw.SJ, hist, npudens, npudist

Examples

# 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
# normal-reference rule-of-thumb, 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 npplot() or into the npudens() function.

bw <- npudensbw(formula=~ordered(year)+gdp, bwmethod="normal-reference")

summary(bw)

## Not run: 

# 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)} rule-of-thumb
# 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, nearest-neighbor bandwidths and a generalized
# kernel estimator for the continuous variable, you would use the
# bwtype="generalized_nn" argument (we override the default tolerances
# to speed things up a bit - don't of course do this in general).

bw <- npudensbw(formula=~ordered(year)+gdp, bwtype = "generalized_nn",
                tol=.1, ftol=.1)

summary(bw)

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# Next, compute bandwidths using likelihood cross-validation, 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 (we override the default tolerances to
# speed things up a bit - don't of course do this in general).

bw <- npudensbw(formula=~ordered(year)+gdp, tol=.1, ftol=.1)

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), tol=.1, ftol=.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
# normal-reference rule-of-thumb, 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 npplot() or into the npudens() function.

bw <- npudensbw(dat=data, bwmethod="normal-reference")

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)} rule-of-thumb
# 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, nearest-neighbor bandwidths and a generalized
# kernel estimator for the continuous variable, you would use the
# bwtype="generalized_nn" argument (we override the default tolerances
# to speed things up a bit - don't of course do this in general).

bw <- npudensbw(dat=data, bwtype = "generalized_nn", tol=.1, ftol=.1)

summary(bw)

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# Next, compute bandwidths using likelihood cross-validation, 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 (we override the default tolerances to
# speed things up a bit - don't of course do this in general).

bw <- npudensbw(dat=data, tol=.1, ftol=.1)

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), tol=.1, ftol=.1)

summary(bw)

detach(Italy)
## End(Not run)