np.condensity: Kernel Conditional Density Estimation with Mixed Data Types

In np: Nonparametric Kernel Smoothing Methods for Mixed Data Types

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.

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.

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) 

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