np.regression: Kernel Regression with Mixed Data Types

npregR Documentation

Kernel Regression with Mixed Data Types

Description

npreg computes a kernel regression estimate of a one (1) dimensional dependent variable on p-variate explanatory data, given a set of evaluation points, training points (consisting of explanatory data and dependent data), and a bandwidth specification using the method of Racine and Li (2004) and Li and Racine (2004). A bandwidth specification can be a rbandwidth object, or a bandwidth vector, bandwidth type and kernel type.

Usage

npreg(bws, ...)

## S3 method for class 'formula'
npreg(bws, data = NULL, newdata = NULL, y.eval =
FALSE, ...)

## S3 method for class 'call'
npreg(bws, ...)

## Default S3 method:
npreg(bws, txdat, tydat, ...)

## S3 method for class 'rbandwidth'
npreg(bws,
      txdat = stop("training data 'txdat' missing"),
      tydat = stop("training data 'tydat' missing"),
      exdat,
      eydat,
      gradients = FALSE,
      residuals = FALSE,
      ...)

Arguments

bws

a bandwidth specification. This can be set as a rbandwidth object returned from an invocation of npregbw, or as a vector of bandwidths, with each element i corresponding to the bandwidth for column i in txdat. If specified as a vector, then additional arguments will need to be supplied as necessary to specify the bandwidth type, kernel types, and so on.

...

additional arguments supplied to specify the regression type, bandwidth type, kernel types, training data, and so on, detailed 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(bws), typically the environment from which npregbw was called.

newdata

An optional data frame in which to look for evaluation data. If omitted, the training data are used.

y.eval

If newdata contains dependent data and y.eval = TRUE, np will compute goodness of fit statistics on these data and return them. Defaults to FALSE.

txdat

a p-variate data frame of explanatory data (training data) used to calculate the regression estimators. Defaults to the training data used to compute the bandwidth object.

tydat

a one (1) dimensional numeric or integer vector of dependent data, each element i corresponding to each observation (row) i of txdat. Defaults to the training data used to compute the bandwidth object.

exdat

a p-variate data frame of points on which the regression will be estimated (evaluation data). By default, evaluation takes place on the data provided by txdat.

eydat

a one (1) dimensional numeric or integer vector of the true values of the dependent variable. Optional, and used only to calculate the true errors.

gradients

a logical value indicating that you want gradients computed and returned in the resulting npregression object. Defaults to FALSE.

residuals

a logical value indicating that you want residuals computed and returned in the resulting npregression object. Defaults to FALSE.

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: first compute the bandwidth object via npregbw and then
    compute the conditional mean:
    
    bw <- npregbw(y~x)
    ghat <- npreg(bw)
    
    Usage 2: alternatively, compute the bandwidth object indirectly:
    
    ghat <- npreg(y~x)
    
    Usage 3: modify the default kernel and order:
    
    ghat <- npreg(y~x, ckertype="epanechnikov", ckerorder=4)

    Usage 4: use the data frame interface rather than the formula
    interface:

    ghat <- npreg(tydat=y, txdat=x, ckertype="epanechnikov", ckerorder=4)
  

npreg implements a variety of methods for regression on multivariate (p-variate) data, the types of which are possibly continuous and/or discrete (unordered, ordered). The approach is based on Li and Racine (2003) 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.

Data contained in the data frame txdat may be a mix of continuous (default), unordered discrete (to be specified in the data frame txdat using factor), and ordered discrete (to be specified in the data frame txdat 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.

The use of compactly supported kernels or the occurrence of small bandwidths can lead to numerical problems for the local linear estimator when computing the locally weighted least squares solution. To overcome this problem we rely on a form or ‘ridging’ proposed by Cheng, Hall, and Titterington (1997), modified so that we solve the problem pointwise rather than globally (i.e. only when it is needed).

Value

npreg returns a npregression object. The generic functions fitted, residuals, se, predict, and gradients, extract (or generate) estimated values, residuals, asymptotic standard errors on estimates, predictions, and gradients, respectively, from the returned object. Furthermore, the functions summary and plot support objects of this type. The returned object has the following components:

eval

evaluation points

mean

estimates of the regression function (conditional mean) at the evaluation points

merr

standard errors of the regression function estimates

grad

estimates of the gradients at each evaluation point

gerr

standard errors of the gradient estimates

resid

if residuals = TRUE, in-sample or out-of-sample residuals where appropriate (or possible)

R2

coefficient of determination (Doksum and Samarov (1995))

MSE

mean squared error

MAE

mean absolute error

MAPE

mean absolute percentage error

CORR

absolute value of Pearson's correlation coefficient

SIGN

fraction of observations where fitted and observed values agree in sign

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 tristen.hayfield@gmail.com, 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.

Cheng, M.-Y. and P. Hall and D.M. Titterington (1997), “On the shrinkage of local linear curve estimators,” Statistics and Computing, 7, 11-17.

Doksum, K. and A. Samarov (1995), “Nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression,” The Annals of Statistics, 23 1443-1473.

Hall, P. and Q. Li and J.S. Racine (2007), “Nonparametric estimation of regression functions in the presence of irrelevant regressors,” The Review of Economics and Statistics, 89, 784-789.

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

Li, Q. and J.S. Racine (2004), “Cross-validated local linear nonparametric regression,” Statistica Sinica, 14, 485-512.

Pagan, A. and A. Ullah (1999), Nonparametric Econometrics, Cambridge University Press.

Racine, J.S. and Q. Li (2004), “Nonparametric estimation of regression functions with both categorical and continuous data,” Journal of Econometrics, 119, 99-130.

Wang, M.C. and J. van Ryzin (1981), “A class of smooth estimators for discrete distributions,” Biometrika, 68, 301-309.

See Also

loess

Examples

## Not run: 
# EXAMPLE 1 (INTERFACE=FORMULA): For this example, we compute a
# bivariate nonparametric regression estimate for Giovanni Baiocchi's
# Italian income panel (see Italy for details)

data("Italy")
attach(Italy)

# First, compute the least-squares cross-validated bandwidths for the
# local constant estimator (default).

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

# Now take these bandwidths and fit the model and gradients

model <- npreg(bws = bw, gradients = TRUE)

summary(model)

# Use plot() to visualize the regression function, add bootstrap
# error bars, and overlay the data on the same plot.

# Note - this may take a minute or two depending on the speed of your
# computer due to bootstrapping being conducted (<ctrl>-C will
# interrupt). Note - nothing will appear in the graphics window until
# all computations are completed (if you use
# plot.errors.method="asymptotic" the figure will instantly appear).

plot(bw, plot.errors.method="bootstrap")
points(ordered(year), gdp, cex=.2, col="red")

detach(Italy)

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we compute a
# bivariate nonparametric regression estimate for Giovanni Baiocchi's
# Italian income panel (see Italy for details)

data("Italy")
attach(Italy)

# First, compute the least-squares cross-validated bandwidths for the
# local constant estimator (default).

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

# Now take these bandwidths and fit the model and gradients

model <- npreg(bws = bw, gradients = TRUE)

summary(model)

# Use plot() to visualize the regression function, add bootstrap
# error bars, and overlay the data on the same plot.

# Note - this may take a minute or two depending on the speed of your
# computer due to bootstrapping being conducted (<ctrl>-C will
# interrupt). Note - nothing will appear in the graphics window until
# all computations are completed (if you use
# plot.errors.method="asymptotic" the figure will instantly appear).

plot(bw, plot.errors.method="bootstrap")
points(ordered(year), gdp, cex=.2, col="red")

detach(Italy)

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# EXAMPLE 2 (INTERFACE=FORMULA): For this example, we compute a local
# linear fit using the AIC_c bandwidth selection criterion. We then plot
# the estimator and its gradient using asymptotic standard errors.

data("cps71")
attach(cps71)

bw <- npregbw(logwage~age, regtype="ll", bwmethod="cv.aic")

# Next, plot the regression function...

plot(bw, plot.errors.method="asymptotic")
points(age, logwage, cex=.2, col="red")

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# Next, plot the derivative...

plot(bw, plot.errors.method="asymptotic", gradient=TRUE)

detach(cps71)

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# EXAMPLE 2 (INTERFACE=DATA FRAME): For this example, we compute a local
# linear fit using the AIC_c bandwidth selection criterion. We then plot
# the estimator and its gradient using asymptotic standard errors.

data("cps71")
attach(cps71)

bw <- npregbw(xdat=age, ydat=logwage, regtype="ll", bwmethod="cv.aic")

# Next, plot the regression function...

plot(bw, plot.errors.method="asymptotic")
points(age, logwage, cex=.2, col="red")

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# Next, plot the derivative...

plot(bw, plot.errors.method="asymptotic", gradient=TRUE)

detach(cps71)

# Sleep for 5 seconds so that we can examine the output...

Sys.sleep(5)

# EXAMPLE 3 (INTERFACE=FORMULA): For this example, we replicate the
# nonparametric regression in Maasoumi, Racine, and Stengos
# (2007) (see oecdpanel for details). Note that X is multivariate
# containing a mix of unordered, ordered, and continuous data types. Note
# - this may take a few minutes depending on the speed of your computer.

data("oecdpanel")
attach(oecdpanel)

bw <- npregbw(formula=growth~
              factor(oecd)+
              factor(year)+
              initgdp+
              popgro+
              inv+
              humancap)

plot(bw, plot.errors.method="asymptotic")

detach(oecdpanel)

# EXAMPLE 3 (INTERFACE=DATA FRAME): For this example, we replicate the
# nonparametric regression in Maasoumi, Racine, and Stengos
# (2007) (see oecdpanel for details). Note that X is multivariate
# containing a mix of unordered, ordered, and continuous data types. Note
# - this may take a few minutes depending on the speed of your computer.

data("oecdpanel")
attach(oecdpanel)

y <- growth
X <- data.frame(factor(oecd), factor(year), initgdp, popgro, inv, humancap)

bw <- npregbw(xdat=X, ydat=y)

plot(bw, plot.errors.method="asymptotic")

detach(oecdpanel)


# EXAMPLE 4 (INTERFACE=FORMULA): Experimental data - the effect of
# vitamin C on tooth growth in guinea pigs
#
# Description:
#
#     The response is the length of odontoblasts (teeth) in each of 10
#     guinea pigs at each of three dose levels of Vitamin C (0.5, 1, and
#     2 mg) with each of two delivery methods (orange juice or ascorbic
#     acid).
#
# Usage:
#
#     ToothGrowth
#
# Format:
#
#     A data frame with 60 observations on 3 variables.
#
#       [,1]  len   numeric  Tooth length
#       [,2]  supp  factor   Supplement type (VC or OJ).
#       [,3]  dose  numeric  Dose in milligrams.

library("datasets")
attach(ToothGrowth)

# Note - in this example, there are six cells. 

bw <- npregbw(formula=len~factor(supp)+ordered(dose))

# Now plot the partial regression surfaces with bootstrapped
# nonparametric confidence bounds

plot(bw, plot.errors.method="bootstrap", plot.errors.type="quantile")

detach(ToothGrowth)

# EXAMPLE 4 (INTERFACE=DATA FRAME): Experimental data - the effect of
# vitamin C on tooth growth in guinea pigs
#
# Description:
#
#     The response is the length of odontoblasts (teeth) in each of 10
#     guinea pigs at each of three dose levels of Vitamin C (0.5, 1, and
#     2 mg) with each of two delivery methods (orange juice or ascorbic
#     acid).
#
# Usage:
#
#     ToothGrowth
#
# Format:
#
#     A data frame with 60 observations on 3 variables.
#
#       [,1]  len   numeric  Tooth length
#       [,2]  supp  factor   Supplement type (VC or OJ).
#       [,3]  dose  numeric  Dose in milligrams.

library("datasets")
attach(ToothGrowth)

# Note - in this example, there are six cells. 

y <- len
X <- data.frame(supp=factor(supp), dose=ordered(dose))

bw <- npregbw(X, y)

# Now plot the partial regression surfaces with bootstrapped
# nonparametric confidence bounds

plot(bw, plot.errors.method="bootstrap", plot.errors.type="quantile")

detach(ToothGrowth)

# EXAMPLE 5 (INTERFACE=FORMULA): a pretty 2-d smoothing example adapted
# from the R mgcv library which was written by Simon N. Wood.

set.seed(12345)

# This function generates a smooth nonlinear DGP

dgp.func <- function(x, z, sx=0.3, sz=0.4)
  { (pi**sx*sz)*(1.2*exp(-(x-0.2)^2/sx^2-(z-0.3)^2/sz^2)+
                 0.8*exp(-(x-0.7)^2/sx^2-(z-0.8)^2/sz^2))
  }

# Generate 500 observations, compute the true DGP (i.e., no noise), 
# then a noisy sample

n <- 500

x <- runif(n)
z <- runif(n)

xs <- seq(0, 1, length=30)
zs <- seq(0, 1, length=30)

X.eval <- data.frame(x=rep(xs, 30), z=rep(zs, rep(30, 30)))

dgp <- matrix(dgp.func(X.eval$x, X.eval$z), 30, 30)

y <- dgp.func(x, z)+rnorm(n)*0.1

# Prepare the screen for output... first, plot the true DGP

split.screen(c(2, 1))

screen(1)

persp(xs, zs, dgp, xlab="x1", ylab="x2", zlab="y", main="True DGP")

# Next, compute a local linear fit and plot that

bw <- npregbw(formula=y~x+z, regtype="ll", bwmethod="cv.aic")
fit <- fitted(npreg(bws=bw, newdata=X.eval))
fit.mat <- matrix(fit, 30, 30)

screen(2)

persp(xs, zs, fit.mat, xlab="x1", ylab="x2", zlab="y",
      main="Local linear estimate")

# EXAMPLE 5 (INTERFACE=DATA FRAME): a pretty 2-d smoothing example
# adapted from the R mgcv library which was written by Simon N. Wood.

set.seed(12345)

# This function generates a smooth nonlinear DGP

dgp.func <- function(x, z, sx=0.3, sz=0.4)
  { (pi**sx*sz)*(1.2*exp(-(x-0.2)^2/sx^2-(z-0.3)^2/sz^2)+
                 0.8*exp(-(x-0.7)^2/sx^2-(z-0.8)^2/sz^2))
  }

# Generate 500 observations, compute the true DGP (i.e., no noise), 
# then a noisy sample

n <- 500

x <- runif(n)
z <- runif(n)

xs <- seq(0, 1, length=30)
zs <- seq(0, 1, length=30)

X <- data.frame(x, z)
X.eval <- data.frame(x=rep(xs, 30), z=rep(zs, rep(30, 30)))

dgp <- matrix(dgp.func(X.eval$x, X.eval$z), 30, 30)

y <- dgp.func(x, z)+rnorm(n)*0.1

# Prepare the screen for output... first, plot the true DGP

split.screen(c(2, 1))

screen(1)

persp(xs, zs, dgp, xlab="x1", ylab="x2", zlab="y", main="True DGP")

# Next, compute a local linear fit and plot that

bw <- npregbw(xdat=X, ydat=y, regtype="ll", bwmethod="cv.aic")
fit <- fitted(npreg(exdat=X.eval, bws=bw))
fit.mat <- matrix(fit, 30, 30)

screen(2)

persp(xs, zs, fit.mat, xlab="x1", ylab="x2", zlab="y",
      main="Local linear estimate")

## End(Not run) 

np documentation built on Oct. 19, 2022, 1:08 a.m.