Kernel Conditional Density Bandwidth Selection with Mixed Data Types

Description

npcdensbw computes a conbandwidth object for estimating the conditional density of a p+q-variate kernel 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 Hall, Racine, and Li (2004).

Usage

npcdensbw(...)

## S3 method for class 'formula':
npcdensbw(formula, data, subset, na.action, call, ...)

## S3 method for class 'NULL':
npcdensbw(xdat = stop("data 'xdat' missing"),
          ydat = stop("data 'ydat' missing"),
          bws, ...)

## S3 method for class 'conbandwidth':
npcdensbw(xdat = stop("data 'xdat' missing"),
          ydat = stop("data 'ydat' missing"),
          bws,
          bandwidth.compute = TRUE,
          fast = FALSE,
          auto = TRUE,
          nmulti,
          remin = TRUE,
          itmax = 10000,
          ftol = 1.19209e-07,
          tol = 1.49012e-08,
          small = 2.22045e-16,
          ...)

## Default S3 method:
npcdensbw(xdat = stop("data 'xdat' missing"),
          ydat = stop("data 'ydat' missing"),
          bws,
          bandwidth.compute = TRUE,
          fast,
          auto,
          nmulti,
          remin,
          itmax,
          ftol,
          tol,
          small,
          bwmethod,
          bwscaling,
          bwtype,
          cxkertype,
          cxkerorder,
          cykertype,
          cykerorder,
          uxkertype,
          uykertype,
          oxkertype,
          oykertype,
          ...)

Arguments

Details

npcdensbw implements a variety of methods for choosing bandwidths for multivariate distributions (p+q-variate) defined over a set of possibly continuous and/or discrete (unordered, ordered) data. The approach is based on Li and Racine (2004) 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.

npcdensbw 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 xdat and ydat parameters. Use of these two interfaces is mutually exclusive.

Data contained in the data frames xdat and ydat may be a mix of continuous (default), unordered discrete (to be specified in the data frames using factor), and ordered discrete (to be specified in the data frames 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 dependent data ~ explanatory data, where dependent data and explanatory data are both series of variables specified by name, separated by the separation character '+'. For example, y1 + y2 ~ x1 + x2 specifies that the bandwidths for the joint distribution of variables y1 and y2 conditioned on x1 and x2 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

npcdensbw returns a conbandwidth object, with the following components:

if bwtype is set to fixed, an object containing bandwidths (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.
The functions predict, summary and plot support objects of type conbandwidth.

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(xdat,ydat)) 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.

Hall, P. and J.S. Racine and Q. Li (2004), “Cross-validation and the estimation of conditional probability densities,” Journal of the American Statistical Association, 99, 1015-1026.

Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.

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 compute the
# likelihood cross-validated bandwidths (default) using a second-order
# Gaussian kernel (default). Note - this may take a minute or two
# depending on the speed of your computer. We override the default
# tolerances for the search method as the objective function is
# well-behaved (don't of course do this in general).

data("Italy")
attach(Italy)

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

# The object bw can be used for further estimation using
# npcdens(), plotting using npplot() etc. Entering the name of
# the object provides useful summary information, and names() will also
# provide useful information.

summary(bw)

# Note - see the example for npudensbw() for multiple illustrations
# of how to change the kernel function, kernel order, and so forth.

detach(Italy)

## Not run: 

# EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we compute the
# likelihood cross-validated bandwidths (default) using a second-order
# Gaussian kernel (default). Note - this may take a minute or two
# depending on the speed of your computer. We override the default
# tolerances for the search method as the objective function is
# well-behaved (don't of course do this in general).

data("Italy")
attach(Italy)

bw <- npcdensbw(xdat=ordered(year), ydat=gdp, tol=.1, ftol=.1)

# The object bw can be used for further estimation using
# npcdens(), plotting using npplot() etc. Entering the name of
# the object provides useful summary information, and names() will also
# provide useful information.

summary(bw)

# Note - see the example for npudensbw() for multiple illustrations
# of how to change the kernel function, kernel order, and so forth.

detach(Italy)
## End(Not run)