np.density.bw: Kernel Density Bandwidth Selection with Mixed Data Types

In np: Nonparametric Kernel Smoothing Methods for Mixed Data Types

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 = 1.490116e-07,
          tol = 1.490116e-04,
          small = 1.490116e-05,
          lbc.dir = 0.5,
          dfc.dir = 3,
          cfac.dir = 2.5*(3.0-sqrt(5)),
          initc.dir = 1.0,
          lbd.dir = 0.1,
          hbd.dir = 1,
          dfac.dir = 0.25*(3.0-sqrt(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,
          ...) 

Arguments

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 as.data.frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which the function is called.
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 NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The (recommended) default is na.omit.
call	the original function call. This is passed internally by np when a bandwidth search has been implied by a call to another function. It is not recommended that the user set this.
dat	a p-variate 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 dat. In either case, the bandwidth supplied will serve as a starting point in the numerical search for optimal bandwidths. If specified as a vector, then additional arguments will need to be supplied as necessary to specify the bandwidth type, kernel types, selection methods, and so on. This can be left unset.
...	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. cv.ml specifies likelihood cross-validation, cv.ls specifies least-squares cross-validation, and normal-reference just computes the ‘rule-of-thumb’ bandwidth h_j using the standard formula h_j = 1.06 \sigma_j n^{-1/(2P+l)}, where \sigma_j is an adaptive measure of spread of the jth continuous variable defined as min(standard deviation, mean absolute deviation/1.4826, interquartile range/1.349), n the number of observations, P the order of the kernel, and l the number of continuous variables. Note that when there exist factors and the normal-reference rule is used, there is zero smoothing of the factors. Defaults to cv.ml.
bwscaling	a logical value that when set to TRUE the supplied bandwidths are interpreted as ‘scale factors’ (c_j), otherwise when the value is FALSE they are interpreted as ‘raw bandwidths’ (h_j for continuous data types, \lambda_j for discrete data types). For continuous data types, c_j and h_j are related by the formula h_j = c_j \sigma_j n^{-1/(2P+l)}, where \sigma_j is an adaptive measure of spread of the jth continuous variable defined as min(standard deviation, mean absolute deviation/1.4826, interquartile range/1.349), n the number of observations, P the order of the kernel, and l the number of continuous variables. For discrete data types, c_j and h_j are related by the formula h_j = c_jn^{-2/(2P+l)}, where here j denotes discrete variable j. Defaults to FALSE.
bwtype	character string used for the continuous variable bandwidth type, specifying the type of bandwidth to compute and return in the bandwidth object. Defaults to fixed. Option summary: fixed: compute fixed bandwidths generalized_nn: compute generalized nearest neighbors adaptive_nn: compute adaptive nearest neighbors
bandwidth.compute	a logical value which specifies whether to do a numerical search for bandwidths or not. If set to FALSE, a bandwidth object will be returned with bandwidths set to those specified in bws. Defaults to TRUE.
ckertype	character string used to specify the continuous kernel type. Can be set as gaussian, epanechnikov, or uniform. Defaults to gaussian.
ckerorder	numeric value specifying kernel order (one of (2,4,6,8)). Kernel order specified along with a uniform continuous kernel type will be ignored. Defaults to 2.
ukertype	character string used to specify the unordered categorical kernel type. Can be set as aitchisonaitken or liracine.
okertype	character string used to specify the ordered categorical kernel type. Can be set as wangvanryzin or liracine.
nmulti	integer number of times to restart the process of finding extrema of the cross-validation function from different (random) initial points.
remin	a logical value which when set as TRUE the search routine restarts from located minima for a minor gain in accuracy. Defaults to TRUE.
itmax	integer number of iterations before failure in the numerical optimization routine. Defaults to 10000.
ftol	fractional tolerance on the value of the cross-validation function evaluated at located minima (of order the machine precision or perhaps slightly larger so as not to be diddled by roundoff). Defaults to 1.490116e-07 (1.0e+01*sqrt(.Machine$double.eps)).
tol	tolerance on the position of located minima of the cross-validation function (tol should generally be no smaller than the square root of your machine's floating point precision). Defaults to 1.490116e-04 (1.0e+04*sqrt(.Machine$double.eps)).
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 10-04 (single precision) or 3x10-8 (double precision)). Defaults to small = 1.490116e-05 (1.0e+03*sqrt(.Machine$double.eps)).
lbc.dir,dfc.dir,cfac.dir,initc.dir	lower bound, chi-square degrees of freedom, stretch factor, and initial non-random values for direction set search for Powell's algorithm for numeric variables. See Details
lbd.dir,hbd.dir,dfac.dir,initd.dir	lower bound, upper bound, stretch factor, and initial non-random 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 non-random initial values for scale factors for numeric variables for Powell's algorithm. See Details
lbd.init, hbd.init, dfac.init	lower bound, upper bound, and non-random 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 TRUE scales lbd.dir, hbd.dir, dfac.dir, and initd.dir by n^{-2/(2P+l)}, n the number of observations, P the order of the kernel, and l the number of numeric variables. See Details

Details

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 (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.

The optimizer invoked for search is Powell's conjugate direction method which requires the setting of (non-random) 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.

Value

npudensbw returns a bandwidth object, with the following components:

bw	bandwidth(s), scale factor(s) or nearest neighbours for the data, dat
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.

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 [e-mail], Jeffrey S. Racine [e-mail]

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

## 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
# 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 plot() or into the npudens() function.

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

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 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.

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
# 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 plot() 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.

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 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.

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) 

np documentation built on March 31, 2023, 9:41 p.m.