R: Kernel Conditional Density Estimation with Mixed Data Types
npcdens
R Documentation
Kernel Conditional Density Estimation with Mixed Data Types
Description
npcdens computes kernel conditional density estimates on
p+q-variate evaluation data, given a set of training data (both
explanatory and dependent) and a bandwidth specification (a
conbandwidth object or a bandwidth vector, bandwidth type, and
kernel type) using the method of Hall, Racine, and Li (2004).
The data may be continuous, discrete (unordered and ordered
factors), or some combination thereof.
Usage
npcdens(bws, ...)
## S3 method for class 'formula'
npcdens(bws, data = NULL, newdata = NULL, ...)
## S3 method for class 'call'
npcdens(bws, ...)
## S3 method for class 'conbandwidth'
npcdens(bws,
txdat = stop("invoked without training data 'txdat'"),
tydat = stop("invoked without training data 'tydat'"),
exdat,
eydat,
gradients = FALSE,
...)
## Default S3 method:
npcdens(bws, txdat, tydat, ...)
Arguments
bws
a bandwidth specification. This can be set as a conbandwidth
object returned from a previous invocation of
npcdensbw, or as a p+q-vector of bandwidths,
with each element i up to i=q corresponding to the
bandwidth for column i in tydat, and each element
i from i=q+1 to i=p+q corresponding to the
bandwidth for column i-q in txdat. If specified as a
vector, then additional arguments will need to be supplied as
necessary to specify the bandwidth type, kernel types, training
data, and so on.
gradients
a logical value specifying whether to return estimates of the
gradients at the evaluation points. Defaults to FALSE.
...
additional arguments supplied to specify the bandwidth type,
kernel types, and so on. This is necessary if you
specify bws as a p+q-vector and not a conbandwidth
object, and you do not desire the default behaviours. To do this,
you may specify any of bwmethod, bwscaling,
bwtype, cxkertype, cxkerorder,
cykertype, cykerorder, uxkertype,
uykertype, oxkertype, oykertype, as described
in npcdensbw.
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(bws), typically the environment from which
npcdensbw was called.
newdata
An optional data frame in which to look for evaluation data. If
omitted, the training data are used.
txdat
a p-variate data frame of sample realizations of explanatory
data (training data). Defaults to the training data used to
compute the bandwidth object.
tydat
a q-variate data frame of sample realizations of dependent
data (training data). Defaults to the training data used to
compute the bandwidth object.
exdat
a p-variate data frame of explanatory data on which
conditional densities will be evaluated. By default,
evaluation takes place on the data provided by txdat.
eydat
a q-variate data frame of dependent data on which conditional
densities will be evaluated. By default,
evaluation takes place on the data provided by tydat.
Details
npcdens implements a variety of methods for estimating
multivariate conditional 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.
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.
Training and evaluation input data 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).
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
npcdens returns a condensity object. The generic
accessor functions fitted, se, and
gradients, extract estimated values, asymptotic standard
errors on estimates, and gradients, respectively, from the returned
object. Furthermore, the functions predict,
summary and plot support objects of both
classes. The returned objects have the following components:
xbw
bandwidth(s), scale factor(s) or nearest neighbours for the
explanatory data, txdat
ybw
bandwidth(s), scale factor(s) or nearest neighbours for the
dependent data, tydat
xeval
the evaluation points of the explanatory data
yeval
the evaluation points of the dependent data
condens
estimates of the conditional density at the evaluation
points
conderr
standard errors of the conditional density
estimates
congrad
if invoked with gradients = TRUE, estimates of
the gradients at the evaluation points
congerr
if invoked with gradients = TRUE, standard
errors of the gradients at the evaluation points
log_likelihood
log likelihood of the conditional density estimate
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.
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
npudens
Examples
## Not run:
# EXAMPLE 1 (INTERFACE=FORMULA): For this example, we load Giovanni
# Baiocchi's Italian GDP panel (see Italy for details), and 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.
data("Italy")
attach(Italy)
# First, compute the bandwidths... note that this may take a minute or
# two depending on the speed of your computer.
bw <- npcdensbw(formula=gdp~ordered(year))
# Next, compute the condensity object...
fhat <- npcdens(bws=bw)
# The object fhat now contains results such as the estimated conditional
# density function (fhat$condens) and so on...
summary(fhat)
# Call the plot() function to visualize the results (<ctrl>-C will
# interrupt on *NIX systems, <esc> will interrupt on MS Windows
# systems).
plot(bw)
detach(Italy)
# EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we load Giovanni
# Baiocchi's Italian GDP panel (see Italy for details), and compute the
# likelihood cross-validated bandwidths (default) using a second-order
# Gaussian kernel (default). Note - this may take a minute or two
# depending on the speed of your computer.
data("Italy")
attach(Italy)
# First, compute the bandwidths... note that this may take a minute or
# two depending on the speed of your computer.
# Note - we cast `X' and `y' as data frames so that plot() can
# automatically grab names (this looks like overkill, but in
# multivariate settings you would do this anyway, so may as well get in
# the habit).
X <- data.frame(year=ordered(year))
y <- data.frame(gdp)
bw <- npcdensbw(xdat=X, ydat=y)
# Next, compute the condensity object...
fhat <- npcdens(bws=bw)
# The object fhat now contains results such as the estimated conditional
# density function (fhat$condens) and so on...
summary(fhat)
# Call the plot() function to visualize the results (<ctrl>-C will
# interrupt on *NIX systems, <esc> will interrupt on MS Windows systems).
plot(bw)
detach(Italy)
# EXAMPLE 2 (INTERFACE=FORMULA): For this example, we load the old
# faithful geyser data from the R `datasets' library and compute the
# conditional density function.
library("datasets")
data("faithful")
attach(faithful)
# Note - this may take a few minutes depending on the speed of your
# computer...
bw <- npcdensbw(formula=eruptions~waiting)
summary(bw)
# Plot the density function (<ctrl>-C will interrupt on *NIX systems,
# <esc> will interrupt on MS Windows systems).
plot(bw)
detach(faithful)
# EXAMPLE 2 (INTERFACE=DATA FRAME): For this example, we load the old
# faithful geyser data from the R `datasets' library and compute the
# conditional density function.
library("datasets")
data("faithful")
attach(faithful)
# Note - this may take a few minutes depending on the speed of your
# computer...
# Note - we cast `X' and `y' as data frames so that plot() can
# automatically grab names (this looks like overkill, but in
# multivariate settings you would do this anyway, so may as well get in
# the habit).
X <- data.frame(waiting)
y <- data.frame(eruptions)
bw <- npcdensbw(xdat=X, ydat=y)
summary(bw)
# Plot the density function (<ctrl>-C will interrupt on *NIX systems,
# <esc> will interrupt on MS Windows systems)
plot(bw)
detach(faithful)
# EXAMPLE 3 (INTERFACE=FORMULA): Replicate the DGP of Klein & Spady
# (1993) (see their description on page 405, pay careful attention to
# footnote 6 on page 405).
set.seed(123)
n <- 1000
# x1 is chi-squared having 3 df truncated at 6 standardized by
# subtracting 2.348 and dividing by 1.511
x <- rchisq(n, df=3)
x1 <- (ifelse(x < 6, x, 6) - 2.348)/1.511
# x2 is normal (0, 1) truncated at +- 2 divided by 0.8796
x <- rnorm(n)
x2 <- ifelse(abs(x) < 2 , x, 2) / 0.8796
# y is 1 if y* > 0, 0 otherwise.
y <- ifelse(x1 + x2 + rnorm(n) > 0, 1, 0)
# Generate data-driven bandwidths (likelihood cross-validation). Note -
# this may take a few minutes depending on the speed of your computer...
bw <- npcdensbw(formula=factor(y)~x1+x2)
# Next, create the evaluation data in order to generate a perspective
# plot
x1.seq <- seq(min(x1), max(x1), length=50)
x2.seq <- seq(min(x2), max(x2), length=50)
X.eval <- expand.grid(x1=x1.seq,x2=x2.seq)
data.eval <- data.frame(y=factor(rep(1, nrow(X.eval))),x1=X.eval[,1],x2=X.eval[,2])
# Now evaluate the conditional probability for y=1 and for the
# evaluation Xs
fit <- fitted(npcdens(bws=bw,newdata=data.eval))
# Finally, coerce the data into a matrix for plotting with persp()
fit.mat <- matrix(fit, 50, 50)
# Generate a perspective plot similar to Figure 2 b of Klein and Spady
# (1993)
persp(x1.seq,
x2.seq,
fit.mat,
col="white",
ticktype="detailed",
expand=0.5,
axes=FALSE,
box=FALSE,
main="Estimated Nonparametric Probability Perspective",
theta=310,
phi=25)
# EXAMPLE 3 (INTERFACE=DATA FRAME): Replicate the DGP of Klein & Spady
# (1993) (see their description on page 405, pay careful attention to
# footnote 6 on page 405).
set.seed(123)
n <- 1000
# x1 is chi-squared having 3 df truncated at 6 standardized by
# subtracting 2.348 and dividing by 1.511
x <- rchisq(n, df=3)
x1 <- (ifelse(x < 6, x, 6) - 2.348)/1.511
# x2 is normal (0, 1) truncated at +- 2 divided by 0.8796
x <- rnorm(n)
x2 <- ifelse(abs(x) < 2 , x, 2) / 0.8796
# y is 1 if y* > 0, 0 otherwise.
y <- ifelse(x1 + x2 + rnorm(n) > 0, 1, 0)
# Create the X matrix
X <- cbind(x1, x2)
# Generate data-driven bandwidths (likelihood cross-validation). Note -
# this may take a few minutes depending on the speed of your computer...
bw <- npcdensbw(xdat=X, ydat=factor(y))
# Next, create the evaluation data in order to generate a perspective
# plot
x1.seq <- seq(min(x1), max(x1), length=50)
x2.seq <- seq(min(x2), max(x2), length=50)
X.eval <- expand.grid(x1=x1.seq,x2=x2.seq)
# Now evaluate the conditional probability for y=1 and for the
# evaluation Xs
fit <- fitted(npcdens(exdat=X.eval,
eydat=factor(rep(1, nrow(X.eval))),
bws=bw))
# Finally, coerce the data into a matrix for plotting with persp()
fit.mat <- matrix(fit, 50, 50)
# Generate a perspective plot similar to Figure 2 b of Klein and Spady
# (1993)
persp(x1.seq,
x2.seq,
fit.mat,
col="white",
ticktype="detailed",
expand=0.5,
axes=FALSE,
box=FALSE,
main="Estimated Nonparametric Probability Perspective",
theta=310,
phi=25)
## End(Not run)