Kernel Conditional Distribution Bandwidth Selection with Mixed Data Types

npcdistbw computes a condbandwidth object for estimating a $p+q$-variate kernel conditional cumulative distribution estimator defined over mixed continuous and discrete (unordered xdat, ordered xdat and ydat) data using either the normal-reference rule-of-thumb or least-squares cross validation method of Li and Racine (2008) and Li, Lin and Racine (2013).

Usage

npcdistbw(...)

# S3 method for class 'formula'
npcdistbw(formula, data, subset, na.action, call, gdata = NULL,...)

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

# S3 method for class 'condbandwidth'
npcdistbw(xdat = stop("data 'xdat' missing"),
          ydat = stop("data 'ydat' missing"),
          gydat = NULL,
          bws,
          bandwidth.compute = TRUE,
          nmulti,
          remin = TRUE,
          itmax = 10000,
          do.full.integral = FALSE,
          ngrid = 100,
          ftol = 1.490116e-07,
          tol = 1.490116e-04,
          small = 1.490116e-05,
          memfac = 500.0,
          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,
          transform.bounds = FALSE,
          invalid.penalty = c("baseline","dbmax"),
          penalty.multiplier = 10,
          ...)

# Default S3 method
npcdistbw(xdat = stop("data 'xdat' missing"),
          ydat = stop("data 'ydat' missing"),
          gydat,
          bws,
          bandwidth.compute = TRUE,
          nmulti,
          remin,
          itmax,
          do.full.integral,
          ngrid,
          ftol,
          tol,
          small,
          memfac,
          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,
          transform.bounds,
          invalid.penalty,
          penalty.multiplier,
          bwmethod,
          bwscaling,
          bwtype,
          cxkertype,
          cxkerorder,
          cykertype,
          cykerorder,
          uxkertype,
          oxkertype,
          oykertype,
          ...)

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.
gdata: a grid of data on which the indicator function for least-squares cross-validation is to be computed (can be the sample or a grid of quantiles).
xdat: a $p$-variate data frame of explanatory data on which bandwidth selection will be performed. The data types may be continuous, discrete (unordered and ordered factors), or some combination thereof.
ydat: a $q$-variate data frame of dependent data on which bandwidth selection will be performed. The data types may be continuous, discrete (ordered factors), or some combination thereof.
gydat: a grid of data on which the indicator function for least-squares cross-validation is to be computed (can be the sample or a grid of quantiles for ydat).
bws: a bandwidth specification. This can be set as a condbandwidth object returned from a previous invocation, or as a $p+q$-vector of bandwidths, with each element $i$ up to $i=q$ corresponding to the bandwidth for column $i$ in ydat, and each element $i$ from $i=q+1$ to $i=p+q$ corresponding to the bandwidth for column $i-q$ in xdat. 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: which method to use to select bandwidths. cv.ls specifies least-squares cross-validation (Li, Lin and Racine (2013), 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 $j$th 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.ls.
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 continuous variable $j$ 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 condbandwidth 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 condbandwidth object will be returned with bandwidths set to those specified in bws. Defaults to TRUE.
cxkertype: character string used to specify the continuous kernel type for xdat. Can be set as gaussian, epanechnikov, or uniform. Defaults to gaussian.
cxkerorder: numeric value specifying kernel order for xdat (one of (2,4,6,8)). Kernel order specified along with a uniform continuous kernel type will be ignored. Defaults to 2.
cykertype: character string used to specify the continuous kernel type for ydat. Can be set as gaussian, epanechnikov, or uniform. Defaults to gaussian.
cykerorder: numeric value specifying kernel order for ydat (one of (2,4,6,8)). Kernel order specified along with a uniform continuous kernel type will be ignored. Defaults to 2.
uxkertype: character string used to specify the unordered categorical kernel type. Can be set as aitchisonaitken or liracine. Defaults to aitchisonaitken.
oxkertype: character string used to specify the ordered categorical kernel type. Can be set as wangvanryzin or liracine. Defaults to liracine.
oykertype: 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.
do.full.integral: a logical value which when set as TRUE evaluates the moment-based integral on the entire sample.
ngrid: integer number of grid points to use when computing the moment-based integral. Defaults to 100.
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
transform.bounds: a logical value that when set to TRUE applies an internal transformation that maps the unconstrained search to the feasible bandwidth domain. Defaults to FALSE.
invalid.penalty: a character string specifying the penalty used when the optimizer encounters invalid bandwidths. "baseline" returns a finite penalty based on a baseline objective; "dbmax" returns DBL\_MAX. Defaults to "baseline".
penalty.multiplier: a numeric multiplier applied to the baseline penalty when invalid.penalty="baseline". Defaults to 10.
memfac: The algorithm to compute the least-squares objective function uses a block-based algorithm to eliminate or minimize redundant kernel evaluations. Due to memory, hardware and software constraints, a maximum block size must be imposed by the algorithm. This block size is roughly equal to memfac*10^5 elements. Empirical tests on modern hardware find that a memfac of around 500 performs well. If you experience out of memory errors, or strange behaviour for large data sets (>100k elements) setting memfac to a lower value may fix the problem.

Details

npcdistbw 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 xdat, ordered xdat and ydat) 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 cumulative distribution at the point $x$. Generalized nearest-neighbor bandwidths change with the point at which the cumulative distribution is estimated, $x$. Fixed bandwidths are constant over the support of $x$.

npcdistbw 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 frame xdat 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 contained in the data frame ydat may be a mix of continuous (default) 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.

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

npcdistbw returns a condbandwidth object, with the following components:

xbw: bandwidth(s), scale factor(s) or nearest neighbours for the explanatory data, xdat
ybw: bandwidth(s), scale factor(s) or nearest neighbours for the dependent data, ydat
fval: objective function value at minimum

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 $k$th 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 condbandwidth.

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.

Li, Q. and J.S. Racine (2008), “Nonparametric estimation of conditional CDF and quantile functions with mixed categorical and continuous data,” Journal of Business and Economic Statistics, 26, 423-434.

Li, Q. and J. Lin and J.S. Racine (2013), “Optimal bandwidth selection for nonparametric conditional distribution and quantile functions”, Journal of Business and Economic Statistics, 31, 57-65.

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.

Author

Tristen Hayfield tristen.hayfield@gmail.com, Jeffrey S. Racine racinej@mcmaster.ca

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 $i$th 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.

Examples

if (FALSE) { # \dontrun{
# EXAMPLE 1 (INTERFACE=FORMULA): For this example, we compute the
# 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.

data("Italy")
attach(Italy)

bw <- npcdistbw(formula=gdp~ordered(year))

# The object bw can be used for further estimation using
# npcdist(), plotting using plot() 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)

# EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we compute the
# 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.

data("Italy")
attach(Italy)

bw <- npcdistbw(xdat=ordered(year), ydat=gdp)

# The object bw can be used for further estimation using npcdist(),
# plotting using plot() 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)
} # }